Usually no but some states (CA) consider them to be equivalent to firearms. A very good full line vendor is Pyramyd Air http://www.pyramydair.com
The best thing to do is download and read Chapman’s book. It will give you the tradeoffs (spring, gas spring, CO2, PCP), what you need for support equipment (from nothing to high pressure air tanks) and what you can do in terms of game.
The gas piston rifles are nice, mild recoil compared with the spring guns. You can get up to about 25 ft lbs at the muzzle. I have a heavily modded TalonSS that gets close to 95 ft lbs. The big bore Korean guns in .45 and .50 push 400 ft. lbs. My favorites are by Air Arms of the UK.