id suggest calling up a gun store or gun range around you, ask them if they rent guns, or if they know of a place that does.
ive been shooting for a couple years now, and recently its become an obsession, its insanely fun. im saving up right now for an ak 47 or 74, i figure an ak should really be in everyones collection if you own multiple guns, if there is one, "best" gun, the ak is it. im going to get a tax stamp and silencer for it, which is gonna be around 1000 to do it right, not including the gun :( but a silenced ak would be the shit.
anyway... id suggest that you buy a legit 22 rifle. a 22mag bolt action would be ideal, it will teach you sooooo much, start with the iron sights (if it has them) learn how to use them extremely well, then start to use a scope, 22 mag will hit somewhat accurately out past 200 yards, and its a very cheap way to learn how to use a rifle effectively and accurately.
id suggest not shooting any animals until you are sure your a good shot. while i dont care much for rodents like rabbits and ground hogs, i never like to see one crawl around for 10 minutes before it dies because of a poorly placed shot. and learn to skin and tan furs, you can use rabbit furs to make some awesome things. and rabbit can taste pretty good too. then you can move up to coyotes, which also have an awesome fur. 22 mag is also much much more effective at killing animals, you can take down a coyote in one decently placed shot, and kill it right away.