I would avoid a Wallyworld special- I've never seen anything in their inventory that was worth beans for ANY kind of air gun. RWS, Gamo, there are a lot of good ones out there. Even Daisy and Beeman have some decent ones, but make sure that is a hunting air gun. Or you can spend a lot on something like a Hammerli. EAA is even bringing in some Russian air guns that are pretty impressive. This is go to the gun shop type stuff most of the time.

I would say, nothing smaller than a .20 caliber, so .20, .22 and .25, but .22 is probably the most common by a lot. Despite what I use, I would put a premium on velocity- I don't know what the recommended threshold is, but I'm going to take a guess at nothing under 700fps or so, just based on what I've exerperinced using subsonic .22LR. But probably the most important factor will be accuracy. If you can get a competition grade air gun that has a high velocity and doesn't use CO2, that might be your best bet. Yes, it will probably be a little more delicate, but the ability to make head shots consistently is worth the expense. They down side is, a good air rifle can start to approach $200 without trying hard- you can spend as much on these as you can a good rimfire rifle or pistol.

And if you have a good air gun, don't feed it junk pellets. The good stuff is a little more expensive, but the weakest part of the system is the part that can make you miss.

EDIT:
As for particular models, looking at what is out there, the Crossman Quest 800 in .22 looks pretty good, at least on paper. Spring powered, light, pretty inexpensive. As does the EAA MP513- one of the Russian imports I was talking about, and they know a lot about air rifle. But I haven't had a chance to get hands on with either.