A good quality pellet rifle is good for small game. I don't know about turkey unless you can stalk or bring them in close enough for a clear head shot. But rats, medium-sized birds, squirrels, and average size rabbits are all possible. You can get a lot of practice with a pellet rifle and do it on short money. $5 gets 500 to a thousand pellets.

I also think your overestimating the price. Yes, you can spend $250 on a pellet rifle but there are nice spring guns, complete with a good scope, that go for about $125. It is entirely possible to set up a temporary 33' indoors range for such a rifle and get in some rainy day shooting. Thirty-three feet, or more, isn't long but the basics of keeping shots in a quarter-inch circle are the same as bigger targets at longer ranges.

It can be fun to make a game of it. Plastic army men and all you can shoot off-hand in a minute as competition. Most of the higher power pellet rifles are single-shot with a muscular cocking process so a minute isn't very long.

For a bit more than $250, I figure, less than $300 I figure, you could get both a nice pellet rifle and a magazine-fed bolt-action .22LR. Both new. Buy used and you might swing it for $250.