For your stated purpose, I think a centerfire is a bit much.
If you're not hunting anything larger than a coyote, there is no need for even a .223. Unless, of course, that's just what you want. If so, then go for it.
A rimfire rifle is going to be easier to shoot, easier on the ears, easier on the wallet, and easier on your shoulder.
Everybody ought to have at least one .22 lr. They are the cheapest and easiest to find and feed.
However, the most fun I have is with my Marlin 917V, .17 HMR. You should be able to shoot MOA with a .17 HMR, right out of the box.
Marlins and Savages are the best, IMO, bolt action rimfires right out of the box for the money.
The CZ line of rimfires are very, very nice, but they'll cost you a little bit more.
The Rugers are nice too, but very expensive for what you're getting.
But, if you're convinced that you want a centerfire, then go with the .223. Again, look at Savage and their budget line Stevens. Very good guns for the money.
Remember, the rifle is only half the equation, you'll need some good glass to go with it. Again, this is where the rimfires have an advantage because you can get nice optics for nice prices. They aren't as violent on scopes as centerfire rifles are.
Happy shopping
ETA: For comparison sake, FMJ .223 goes anywhere from $7 to $12 per 20. Whereas .17 HMR and .22 WMR (.22 Mag) go for about $12-$14 per 50.
The ever popular .22 lr goes for considerably less. $18 for 500. It probably counts for more rabbits on the table than everything else combined.