This is kind of a polarizing thing to do but hey, if you happen to be going to Vegas, you love the 2nd amendment, you choose to bring your choice of firearm and decide you'd like to get a Nevada concealed weapons permit, it's something to do and you get to do some shooting at one of the most incredible ranges in the country.
First things first. As of spring 2012, Nevada is an open carry state, meaning it is legal to carry a weapon on your person visibly state-wide without any special permit provided you're a legal owner of a weapon and would not fail any of the requirements to purchase another one. Of course you don't want to do this in Las Vegas because people will freak out and you'll get arrested for disturbing the peace or any number of other charges even though the actual carrying of the weapon was legal; so, I repeat, don't do that. Also, open carry is not concealed carry, do not conceal a weapon on your person or in your car if you do not have the permit. In fact, in Las Vegas proper, you should not even have the weapon loaded or the magazine in the gun in your car as Metro PD have been known to arrest people for that, in violation of state law, but who wants to deal with fighting that.
Okay, with that out of the way, Nevada's concealed carry permit process involves a classroom session taught by an approved instructor and shooting of your own weapon to demonstrate competency. You can find a lot of places offering these classes on weekends, the one I took was with instructor Tim Brown of SWIFT. He's an NRA certified instructor, phone number is 702-278-9155. He does these classes for FUN, and keeps them very small, so they're very good. I took the class in May 2012 with six other people. Don't forget you can check a gun in your luggage; I flew in from Florida with my gun. If you want just the Nevada permit, he charges $75, he can also do Nevada and Utah combined for a slightly higher fee. He teaches the classes on Sunday mornings around 8am at the Clark County Shooting Complex:
If you want to do it, give him a call to reserve a seat in his next class. Show up on that day with $75, your OWN gun and 50 rounds of ammo. Nevada will issue a permit for semi-auto and/or revolver; if you want the ability to carry both, you need to bring both and shoot both; you cannot get the permit to carry a revolver if you don't test with one, and vice versa. You also cannot take the class or get a permit if you do not own and bring your own gun to the class; no borrowing a friend's, etc.
After the class, test and shooting test, you'll get all your paperwork that has to be presented to the Clark County Sherrif on a week day along with $97.50 and you'll get a five year permit mailed to you a couple months later. It's pretty easy.
With that out of the way, check out the Clark County Shooting Complex. It is ridiculous. They have ranges for pistols, shot guns, archery, rifles, regular targets, steel targets, educational programs, etc. There is an on-site cafeteria and an 80-slot RV park for people staying over. They have night shooting, a pro shop, storage, ammo, clays and trap shooting, etc. The place is huge, very well kept and like no other shooting range.
It is NOT close to the strip, so allow enough time to get there if you're taking a Sunday class; allow 40 minutes from Caesars Palace for example. Here's an idea of how far it is:
This is out on the 100 yard pistol range: