Activity requirements
You have to be over 18 years of age to shoot in a range and they'll make you sign a health and safety waiver covering a variety of medical conditions, other then that, it doesn't really take much (some places will have an instructional video and small written exam though).

Cost
I've been twice and the first time I went I paid by the box of ammo, and they make you buy theirs so it's not cheap. The second time we selected 3 machine guns and fired them down the range with a prepaid amount of ammo which worked out a bit better. If you're going to a tourist destination such as Vegas you're going to get a bit ripped off; there's certainly other ways to fire a gun, but if you're going to do it in a range, try to get out of town a bit.

What Fiked recommends
Personally, the M249 SAW Machine Gun does it for me. The more modern the gun, the more accurate they are and the less painful recoil they produce. If it's your first time firing a gun, start with a small handgun and work your way up to old shotguns and machine guns. If you're not firing in a range, make sure you bring hearing and eye protection with you.

Best time to do this
There's no real off season for shooting a gun, but different ranges tend to have days where there's less customers and have promotions to try and get people through the door. Have a look for places in your area and see what they've got going on.