.22 airgun is legal for turkey, but .22 rimfire isn't. Odd, but true out here.
okay, powerplants, and budget.
If you can spend close to $400, the discovery in .22 with pump is THE way to go. flat out. Well built, VERY high powered, shoots just as well at 30F as 110F, accurate, and can get decent ballistics with even the heavy (and expensive!) 26 grain pellets.
If you want a piston gun, I'd highly recommend at this point getting a gas piston instead of spring piston. The Walther Falcon, with gas ram, is about $350. Currently, a refurbished one on one site is $175 with spring piston. (that's .22 and will definitely get the bunny, turkey, or immortal squirrel)
I'm not going to say anything against the decent Crosman Quest for a budget springer, but you can't leave them cocked for extended periods of time and if you are seriously wanting to hunt, you may need to sit in a hide for an hour or more. I own one in .177 and it's good enough- it's been tuned by flying dragon and that helps some. It gets real world 852FPS with hunting (10grain) pellets.
CO2- the 2260 from crosman is possibly the best stock rifle for under $100. HOWEVER, temperature has a huge effect on the performance of CO2.
I like pumpers. While not the most powerful much of the time, they can and do deliver in all weather, with no accessories. While the venerable benjamin is a nicely built gun, It's heavy. and big. and heavy.
My preference for a pumper is to get a lightly customized crosman based .22 - the 1377 pistol (which is powerful enough for close range hunting!) is used as a base, a .22 barrel, generally 14 or 18 inches is added. Various people have various ideas about modifying the valve and pumping system. I expanded and re angle the port on my valve and lightened the spring and with a 14 inch barrel I get 14.3 grains of .22 at 650fps on 14 pumps. All very basic mods, and the grip set with integral stock isn't hard to remove for backpacking. (In canuckia, you can get the base model called a 2289, which will give you about 150FPS less)
There used to be several very good lightweight multipump hunting rifles out there, they are all used at this point, and out of production. If I was shopping the market, I'd choose a crosman 140 or 101, and have an airgunsmith give it a once over (or just buy it from an airgunsmith, as I did with my 101)
If you want oldengear, that 101.... man, that 101! feels like and old time .22 rimfire, has a very usable tombstone peep sight system, and while they vary from gun to gun, mine gets 570FPS on 8 pumps with standard crsoman premier 14.3 grain pellets. I get 535 with the heavier crow magnums on 9 pumps. I can easily hit an empty pellet tin at 30 yards with it.
I'd take the 392 over any spring piston airgun, but she's heavy. If you get one, I might still have a spare peep sight laying around.
Caliber- I prefer .22 - .177 in a high speed gun can acupuncture game and leave it running off wounded. .22 tends to not do this. .177 pellets are going to generally be more effected by wind and range factors. They are cheaper, and the crosman heavies (10+ grains) make a decent "100fps" classe .117 into a good medium range hunter. They will work against squirrel in a 1377 at 10 yards quite easily, too- though a 100% dead clean kill is hard on the Immortal Squirrel. You may have to put in a finishing shot.
.22 is a bit more expensive but generally just performs better for hunting. I prefer it for birds, as well. the standard 14.3 grain crosman premier pellet is generally going to be accurate and hard hitting enough for anything except specialized shooting.
Now, costs of ammo- I have and use beeman crow magtnums in both calibers, but sparingly. It's more expensive than shooting bulk box federal .22LR. For me, if it's costing more than a .22 its' costing too much, and I stick about 90% with the .177 crosman premier heavies and the .22 crosman premiers. If you find a decent, inexpensive wadcutter that's very accurate, use it by all means- especially in a .177- better stopping power.
(I'm even considering selling a few of the pumpers off)