Airguns are very much an option for hunting. I have a Beeman R9 I use at my ranch for pest control. It'll definitely do the job on rabbits and squirrels out to 50-60 yards with one shot.
Spring piston is the way to go in this situation. The Sheridan/Benjamin Silver/Blue Streaks were good guns in their day (I have a real early Silver Streak). But they require many, many pumps to get up to hunting strength. Precharged pneumatics charge off SCUBA or special PCP tanks at 3000 to 4500 psi. They come in calibers up to .50, and can drop a bull elk at 100 yards. But they require a lot of peripheral gear and maintenance. A springer uses one stroke to cock a spring (sometimes an air piston) that can propel a pellet at around 1000 fps. Mine's .177, which is easier to find ammo for in wide varieties. For heavier hunting you can go to .20, .22, or even .25. But you tend to lose distance and speed with larger, heavier pellets.
Good quality pellets make a huge difference, too. I use hollow points or wadcutters when hunting. Every gun has it's own preference on pellets. They all shoot differently.
Shooting a springer is different from shooting a firearm, too. It recoils in both directions. Something to bear in mind when buying a scope. A springer will destroy a firearm scope very quickly. You have to use a scope designed for airguns. You also have to hold the gun differently. Don't try to prevent it from recoiling. Hold it lightly and let it spring back and forth naturally. Also, you have to practice. A body shot with an airgun is usually a bad thing. A wound that may or may not be fatal after a long period of suffering. If you can't do a head shot with reasonable confidence, don't take the shot. Basically you should be able to hit a nickel at 50 yards or you shouldn't be hunting with it.
Expect to spend real money on a real gun. The stuff they sell at WalMart, K-Mart, etc. is crap. It's OK for kids punching paper with BB's if they don't care about accuracy, but they're not real hunting weapons. Don't believe the hype about muzzle velocity, either. If it's not complete BS, it's measured with the lightest possible pellets that will travel in some random direction. FPE (foot pounds energy) is the only real way to compare relative power of the guns.
These guys have a good selection of "adult airguns." I've had some minor dealings with them and had good luck:
StraightshootersKevin B.