Using anything more powerful than a .22 magnum will result in your rabbit being gutted, skinned, and shredded into burger, what little will be left. My .223 would "literally" disintegrate Rock Chucks (a relative of the ground hog/wood chuck).
My all-time favorite for rabbits was a double-barrel .410. Good for birds at reasonable range also.
My personal favorite is 22 mag but if your are just plinking then nothing beats the good ol' 22lr.
Anything other than .22lr, .22mag, .17 is too much for small game.
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.
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.
Well, I guess I need to modify my first response. I have never shot the .17HMR but I had been under the assumption that it was center fire cartridge.
Shows what assumption will get you!
Listen to zacii, he's obviously more astute that I am!
That's too light for deer or larger game, IMHO.
how much is a box of 50 HMRs?
Depending on where you are hunting a .22 LR/22 mag /.17 HMR may be too much due to the bullet travel as well. Shotguns are used for most birds and some other small game due to the bullet travel . Pellets lose momentum faster and have a pattern of shot making hits easier. Knowing your backstop is still important. Even for deer in some areas shotguns are used due to population density .
what do you intend to hunt?
where will this be location wise?
The op might be best served with a good old Remington 870 Express, especially if he is just getting into hunting. It is one gun that can fill several roles and is a great starter for hunting. Remington also offers a combo deal most every fall that includes a screw choke barrel and a rifled slug barrel. There is usually a rebate too. This is the way I started hunting years ago and it allowed me to try different game without having to buy many guns.
Just get a 22 magnum and be done with it. You can shoot WRF in it which are heavier thab the 22 lr, but less power than the WMR. And if you need to go after coyote sized animals you can use the WMR. Great versatility.
Or, if you want a light centerfire, go with the 22 Hornet, which extends the range from the 22 WMR to about 100 yards more, depending on the quarry.
Anything 223 or larger is way too much power for bunnies and birds.