trg42wraglefragle, you will be what's known as a "non-immigrant alien". Under federal law must have a valid hunting license in order to be in possession of firearms or ammunition. Even if you go to a range that rents guns, or someone takes you shooting, you still need that. If you're not actually going to go hunting, then it doesn't matter what state your hunting license is from. Some states will sell them to you online.
Once you have a hunting license, in order to buy a gun, you'll need the following:
-utility bills proving 90 days of residency (don't bother trying until you've been in the country for more than 90 continuous days, you will be denied at the background check)
-your immigration documentation (probably an I-94 form stapled into your passport) with your alien/admission number
-a driver's license for the state you're in
-your social security number (optional but highly recommended)
-a firearms dealer who won't freak out at the idea of having to fill in the form a little differently (these can be hard to find)
You will need to write your alien number on the form (there's a section specially for that). If they call in the background check in front of you, listen carefully - if they get stuck on the alien number, i.e. you hear the dealer repeating it to the NICS agent again and again, it usually means they have missed the fact that you're a non-immigrant alien. There's a question on the form specifically for that but they often overlook it because they'll expect you to have answered "no". I've had to point this out while they're on the phone a few times and it has always solved the problem.
Finally, don't be surprised if you get denied the first time. You can appeal but it takes quite a while. I would simply try again a month later if I were you.