Which gun show in the US would be best for a tourist to buy a gun?

You can always find a way to buy a gun. This is never going to be a problem.

You cannot carry it around with you though, as a non-resident and non-citizen you cannot get a firearm permit (a carry permit).

Now, you can use a firearm? Sure - walk into any gun range and have fun with all kinds of weapons available there.

As far as a hunting license goes each state has its own regulations and the US federal law states that the following are the exceptions to those that can "own or possess" (note: doesn't say buy) firearms:

non-immigrant aliens admitted into the United States for lawful hunting or sporting purposes or if the non-immigrant alien falls under one of the following exceptions:

• possesses a valid hunting license or permit issued by any US state

This means that you have to already have obtained a hunting permit. To obtain a hunting permit, you have to check with each state as the requirements vary greatly.

Some quick googling resulted in the following:

Texas:

A RESIDENT is a person who has lived continuously in Texas for more than six months immediately before applying for a license. Members of the United States Armed Forces (and their dependents) on active duty anywhere are entitled to purchase a resident license, but the Texas Resident Active Duty Military “Super Combo” package is available to Texas residents only.

California:

Fish and Game Code Section 70 defines "Resident" as any person who has resided continuously in the State of California for six months or more immediately prior to the date of their application for a license or permit, any person on active military duty with the Armed Forces of the United States or auxiliary branch thereof, or any person enrolled in the Job Corps established pursuant to Section 2883 of Title 29 of the United States Code.

Florida:

For the purposes of hunting or fishing in Florida, a Resident is defined as: Any person who has declared Florida as his or her only state of residence as evidenced by a valid Florida driver license or identification card with both a Florida address and a Florida residency verified by the Department of Highway Safety; or Any member of the United States Armed Forces who is stationed in Florida (includes spouse and dependent children residing in the household).

So, it would be important to check with the state where you plan on using the firearm.

Note that hunting is usually restricted, and a hunting permit does not (normally) grant you the right to carry a weapon (sometimes called License to Carry (LTC)).

In a word, none, as you would not be eligible to purchase a firearm. As a matter of fact, the Federal form required to purchase a firearm establishes US citizenship, Social Security number, etc., and is electronically transmitted to the Federal Bureau of Investigation (FBI) for instant verification. Vendors might not even show you a weapon if they don't believe you would be eligible to purchase.

The relevant section of the US Department of Justice Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF) Federal Firearms Regulations (page 215):

[18 U.S.C. 921(a)(33); 27 CFR 478.11]
R. NONIMMIGRANT ALIENS
(R1) Who is a nonimmigrant alien? A nonimmigrant alien is an alien in the United States in a nonimmigrant classification as defined by section 101(a)(15) of the Immigration and Nationality Act (8 U.S.C. 1101(a)(15). Generally, “nonimmigrant aliens” are tourists, students, business travelers, and temporary workers who enter the U.S. for fixed periods of time; they are lawfully admitted aliens who are not lawful permanent residents.

[27 CFR 478.11]
(R2) May a nonimmigrant alien who has been admitted to the United States under a nonimmigrant visa possess a firearm or ammunition in the United States? An alien admitted to the United States under a nonimmigrant visa is prohibited from shipping, transporting, receiving, or possessing a firearm or ammunition unless the alien falls within one of the exceptions provided in 18 U.S.C. 922(y) (2), such as: a valid hunting license or permit, admitted for lawful hunting or sporting purposes, certain official representatives of a foreign government, or a foreign law enforcement officer of a friendly foreign government entering the United States on official law enforcement business.