Your best bet if you want to fire a gun is not a gun store, but a shooting range.

Most shooting ranges are clubs or semi-private, and won't rent guns to individuals who just walk in. What you're looking for is a tourist-oriented shooting range such as this one. These ranges tend to be found in the more tourist-oriented parts of the western states (eg. Las Vegas), and usually emphasize exotic or famous guns (such as the Tommy gun or the Colt Single Action Army).

If it's advertising "machine guns", it's probably the sort of range you're looking for. And no, you almost certainly won't be able to fire an actual machine gun. What they usually have are automatic rifles and submachine guns.

Nevada: We have businesses that cater to this desire. There are multiple businesses here in Las Vegas that offer this sort of service with AFIAK minimal paperwork. As they are dealing with plenty of total novices there is obviously a lot of supervision.

Multiple such businesses advertise about being able to shoot machine guns. There are also places that allow shooting of grenade launchers--but with inert rounds, they don't go boom. If allowed at all minors are generally only permitted the light weapons and with the places that have them, pintle-mounted weapons. (If the gun is mounted the recoil is irrelevant.)

I have never attempted to count them up but if you spend any time here you'll see ads.

There are also more traditional gun ranges that rent guns, rent lanes and sell ammunition. I wouldn't suggest this route for the total novice, though.

Note that at least traditional gun ranges often will not rent a gun to a lone individual unless they have a concealed carry permit, I have no idea if this also applies to the places catering to tourists. (This is not a matter of law, but rather there have been some cases where someone appears to be an ordinary customer but whose real objective is suicide. Gun ranges very much don't like it when that happens.)

Legally, what's going on here is that so long as the gun stays on the premises the law considers it to be in the possession of the business. The business needs to comply with all the paperwork (which for the machine guns and grenade launchers is considerable), the customer does not. Explosive rounds are a whole another level of paperwork, I don't even know how it's handled but some mechanism must exist as there are fully operational field artillery pieces in private hands--they are used for avalanche control. 8 hours ago I was looking at a warning sign showing what an unexploded round looks like and saying that if you find one to leave it alone and report it. I've never heard of anyone actually encountering one, though--I was simply at the point where the trail crossed into the canyon containing the ski resort. (I wasn't worried about being blown up. It's just there had been a recent snowfall that obscured that part of the trail and I figured it had to be pretty close to the sign.)

California, North Carolina and Florida are states where I have shot a gun at a gun range, as a walk-in, using international ID. Pennsylvania should be similarly easy.

New Jersey I was told I had to take a friend with me, but had nobody to go with. That was a safety thing and I believe a policy of the range, not a law.

New York I didn't even bother finding out.

I would highly recommend going with someone who has done it before, or paying for a professional from the range to get you started. A good range will have a range officer to assist shooters.

Don't go in your own pretending you know what you're doing when you haven't. You'll hurt yourself, or someone else. There's no shame in telling the range officer you're new and want help. In my experience they are very good and customer focused.

Certainly in Hawaii they can. Indeed on Kalakaua Ave. in Honolulu there are plenty of people distributing flyers for gun ranges to Japanese tourists.

I do not know how to find such publicly accessible gun ranges in general, but in some touristy areas they are advertised openly.

I took a UK citizen to a turkey shoot years ago because he mentioned he had never fired a gun...

no one asked for his ID.

this is a big part of American life

That actually depends on who you ask.

I know a number of US citizens that have never fired a gun and are perhaps horrified at the thought of doing so. While that is odd to me I would not say that they were missing a big part of American life.

Here's my advice:

1. 
   

   First figure out what you want to shoot.

   
   
   

   • Shotguns - less regulated than other guns in the UK according to a friend of mine... (the one I took to the turkey shoot) so I assume that isn't what you're after. Plus they're less interesting.

   
   

   • Handguns:
     
     There are a lot of places that can accommodate this.
     
     Get a lesson with something small (low kick) like a 22 or 9 mm.
     
     Learn to hit a head sized target at 10 feet (~3m).
     
     Then try to hit a body sized target at 15 feet (~5m).
     
     You'll learn that the 20+ foot (6+ meter) shots you see in TV and movies are very improbable with a short barrel weapon.
     Then move up to something that has a lot of kick like a 44 or 45, and try to hit the target at 10 feet - just for the experience.

   
   

   • Machine guns:
     
     This is a lot more expensive, but could be super cool.
     
     It will ruin movies for you - the next time you see the hero or big bad guy pick up a 60-cal and shoot it for a minute straight (possibly one handed) you'll be able to tell your friends back home, "No way someone could do that, here's several reasons why..."

   
   


2. 
   

   Before you come, communicate with a gun range (not a gun store) to find out what is available there. Do this with multiple places, prices and services can vary a lot from place to place.
   
   Bonus points to sites that mention beginner lessons - that's you.

   
   
   
   • If you want to fire a machine gun - most gun ranges I'm familiar with do not have this capability
   
   
   • • Clear this in advance with the range - the person who owns the machine gun has to have a Federal Class A fire-arms permit (at a minimum, local laws may require more).
     
     
   
   
   • • Make sure there is more than one person who can help you do this.
       
       If it is just one person, that person may be sick that day - spoiling the fun.
     
     
   
   
   • If you want to fire a high powered rifle (like a sniper) make sure they can accommodate.
   
   
   • Regular handguns (9 mm, 40, 44, 45) can be shot in most indoor ranges (so you don't have to worry about weather).
     
     Indoor ranges are designed and built to handle a specific power of weapon.
   
   

I'm sorry I can't say for sure, but I don't believe it will be a legal issue for you.

I hope this post helps you with your planning.

I went to a shooting range in Philadelphia with my mother (I had J1 visa, she on tourist visa from Europe) and it was no issue whatsoever. Just bring your passport.

As previously mentioned, you are looking for a shooting range or gun club if you are not visiting someone who already has firearms. A local example of this is Scottsdale Gun Club in Arizona. According to their FAQ's:

I AM FROM A DIFFERENT STATE/COUNTRY. CAN I STILL SHOOT AT THE SCOTTSDALE GUN CLUB?
Yes! All you need is a valid (government issued) photo ID in order to use our range (Passport, ID card, Driver’s License, etc).

In addition, gun clubs rent handguns and long guns and have ammunition available for purchase.

but as tourists this is a big part of American life to try and understand a bit better

Just as a caveat, most of the other answers are featuring gun ranges with esoteric guns. In the US, machine (fully automatic, shoots as long as the trigger is held) guns are highly restricted (AK47, Tommy Gun, P90, etc), and thus you have gun ranges that specialize in these more exotic guns that your average American does not have access to. The vast majority of privately owned guns are semi-automatic (the gun shoots one bullet per trigger pull). If you're looking for an average gun (like the infamous AR-15), these are much more common and many gun ranges will gladly rent them to you.

Your best bet in permissive states would be to find one that has Constitutional Carry as its controlling gun law philosophy, with Shall Issue concealed carry as a close second (which covers most states). No state may ban firearm ownership (and, by proxy, gun ranges), but some states have restrictive permits and use the permitting process to limit legal gun ownership and usage. For instance, the state of New York requires you to have a pistol permit. New York City requires you to have a permit from the city itself. I would not go there as a tourist looking to shoot a gun. Neighboring New Hampshire, however, is a Constitutional Carry state.

Virtually anywhere in America and Canada you can visit a range and they will accommodate you. Be prepared: it's relatively expensive (can be up to $100USD or more depending on your intended stay).

Simply google: gun range [insert area]

If you have specific interests, such as a guided tour in rural areas, you will have to specify these.

I'm surprised you have not visited any ranges in the UK. Visiting a friend in Wales, we went to the North Wales Shooting School and it was not a problem at all for me to use their range, so long as I had ID.

Laws about renting firearms and range time will vary from state to state. I would search for a gun range in a city that you plan to visit. Then, contact them directly! (In Arizona and Nevada, there are businesses that cater to tourists)

Explain that you want the experience of safe shooting. They should be able to accommodate you (if possible).
Supervised possession (and shooting) by a non-US citizen is generally allowed. Shooting a small caliber (diameter) rifle or pistol is the same as shooting a large one (just less recoil/pain).

Background: I am from the UK with very little gun experience. This may well be specific to Florida, but I went to https://machinegunamericaorlando.com/ and was able to shoot M16, Ak-47, MP40 and a Colt pistol (could chose from a range of weapons). Can't even remember if I needed my passport (possibly did) but did not require anything else (I'm over 25 in case this makes a difference). We shot, one at a time, with an instructor at our shoulders at all times so felt very safe.

It certainly was not cheap though (especially when factoring in the [infamous] US tipping for the instructor), but for someone who has a mild interest in shooting guns (guessing from the OP sounds similar to my level of interest) it was an enjoyable experience.

I can add knowledge of the state of Indiana to the rest of the answers; I visited friends in South Bend some ten years ago and went to a shooting range with them and fired their three handguns. I am not entirely sure that I had to show my passport, but I think so. I think my friends were members of the club. There was no hassle at all about us bringing a twelve year old child, and her Swedish father let her even try to shoot so I am guessing there were no restrictions about that.

We shot at circles and squares but there were human silhouettes available, which I discovered when a guy came in to teach his girlfriend to shoot. That made me really queasy and took the fun out of the rest of the stay at the range.

Yes, you may rent and shoot a gun in New Hampshire which recognizes constitutional carry. You do not need to be a US citizen. You may also rent a machine gun to shoot for $75 (though the bullets might cost $0.25 each and add up fast). For example:

http://www.gunsnh.com/faq

I am not a NH resident. Can I still shoot there?
Yes, as long as you meet the age and ID requirements.You must be at least 18, or accompanied by a parent or guardian who is at least 21. Everyone who is 18 or older must present government-issued photo ID or a Passport to go onto the range.

They have restrictions on renting alone. Read the entire rules of wherever you go and call to double check. They also have classes you can arrange.

Manchester Firing Line:
https://goo.gl/maps/CGdJ5NJYtcCvqifcA

Arizona is another constitutional carry state. Here is a list of US states with constitutional carry:

https://en.wikipedia.org/wiki/Constitutional_carry#U.S._jurisdictions_that_have_constitutional_carry

I cannot speak for every state in the Union, but come on up to Jefferson. "We'll sort ya out."
Now that I've haphazardly opened with an attempt at humor, (Jefferson State in northern CA is a state of mind in its current form, not an actual state fyi)
California does indeed have strict firearms statutes,
and they are enforced.
But if all you desired was to hit the range and try out everything from Classic Western Cinema, to darn near John
Wick, any number of indoor ranges would be more than happy to assist you in your endeavor to experience "American gun culture"
Northern CA, or South.
The bay area is....well lets just say that SF is a great place, but known for its landmarks- history -people and food.
Not gun ranges.
I suggest looking at some Yelp feedback concerning the ranges you want to try, and then call them.
The rest will be very easy once you explain you are from the UK, and you want the 2nd amendment experience (I exaggerate only slighty here)
They will be more than happy to take your business,
and show you a good time.
Safely of course.
If money isn't a large budgetary concern, you might even just try and hire a range officer for the day.
I can personally assure you this:
No matter where you visit for your range day, CA or a less restricted state like Nevada or Texas, you will leave with some different ideas about this subject and hopefully a big smile.
Have fun shootin!

In general, the right to keep in bear arms exists for all people in the United States, not simply citizens, so at a federal level, there is no bar (provided you enter the country legally -- federal law bars people in the USA illegally from possessing firearms.)

Some people have mentioned that state laws concerning ownership and possession of firearms vary greatly. This is true, but those laws tend to impact the ease by which citizens or resident aliens might go about buying, possessing, or carrying those firearms for self-defense purposes, or using them for hunting or other sporting purposes. For someone who simply wants to come in to the country, go to a range, and shoot a handgun or rifle for an hour under supervision, there generally won't be much of an issue.

I would be more concerned with availability of ranges. In some cities in some locations, there are few gun ranges within the city limits. New York City or Chicago are classic example, so (oddly) Detroit. Generally cities in the South, Midwest, or Rocky Mountains will likely have more such facilities. (As does Las Vegas). In general, it is my experience that even if you go to a large city in a state with more restrictive gun laws (like, say, California) there will be ranges who would be willing to cater to your desires. (It is undisputable in American jurisprudence that the Constitution protects an individual right of the "people" (not just citizens) to keep and bear arms, and while states may attach various regulations on that right, they cannot stop people from possessing firearms.

A lot of times, gun ranges are mentioned/reviewed on Yelp. I encourage you to call a few people in the place of your final destination in America and ask some questions.

As with any activity involving things that go "bang" don't forget that your first goal is to come home safely. Pay attention to all range safety rules, and especially the "four rules" https://www.itstactical.com/warcom/firearms/colonel-jeff-cooper-carry-conditions-and-firearm-safety/

This is federally illegal, unless your country is part of the Visa Waiver Program or you have a U.S. hunting license.

18 USC § 922(g)(5)(B) states:

(g) It shall be unlawful for any person—

(5) who, being an alien—

(B) except as provided in subsection (y)(2), has been admitted to the United States under a nonimmigrant visa...

If you have been admitted to the United States under a nonimmigrant visa, it is a federal offense to possess a firearm. The relevant exception in section (y)(2) is admission for "sporting purposes" (that is, you entered specifically to hunt or to attend a shooting competition) or having a hunting license.

The ATF confirms that this does not apply to countries that do not require visas:

Does the prohibition on the receipt and possession of firearms and ammunition by aliens in nonimmigrant visa status apply to nonimmigrant aliens who lawfully enter the United States without a visa?

No. A nonimmigrant alien who is lawfully admitted to the United States without a visa (e.g. Visa Waiver Program), may acquire or possess a firearm in the United States.

Even if you have one of the above exceptions, beware that no party may legally permanently transfer a firearm to you.

18 USC § 922(b)(3) states:

(b) It shall be unlawful for any licensed importer, licensed manufacturer, licensed dealer, or licensed collector to sell or deliver—

(3) any firearm to any person who the licensee knows or has reasonable cause to believe does not reside in...the State in which the licensee’s place of business is located, except that this paragraph (A) shall not apply to the sale or delivery of any rifle or shotgun to a resident of a State other than a State in which the licensee’s place of business is located...and (B) shall not apply to the loan or rental of a firearm to any person for temporary use for lawful sporting purposes

Section (a)(3) says the same of private parties. Because you do not reside in any State, exception (A) does not apply to you, and a party may transfer a firearm to you only under exception (B), i.e. temporarily for lawful sporting purposes.

There is also section (a)(9), which prohibits people who do not reside in any state from receiving firearms other than for lawful sporting purposes. This should not affect you, as a shooting range definitely qualifies as lawful sporting purposes.

...can we easily visit a range to have the experience of shooting?

tl;dr

Yes

How much more is there to say? Almost anywhere in USA you can find shooting ranges online. Call to ask if they'll welcome visitors from UK. Almost any range will have quite a few pistols and rifles from which you can choose. They'll provide ammunition, targets, earmuffs, supervision, and a little training.