A lot of people like a nice accurate rimfire, me included. Anschutz ia a great choice, but before you buy take a look at the new Weatherby MK XXII. You can get it in .22 LR or 17 HMR if you prefer. Either way you're getting a Anschutz barrel and action, it just has a much nicer stock and comes with swivel studs installed so you don't have to worry about doing it yourself and possibly screwing up the stock.
As far as calibers go, it depends on what you want to do with it. I have one in .22 LR and it will absolutely kill a rabbit or any other small game way beyond 30 yards. I consider 75 yards to be the outer limit for hunting accuracy and killing power, but it will reach out well beyond that with practice and well placed shots. I also have an Anschutz 1517 MPR in 17 HMR that I use sometimes to reach out to 150 yards max for small game. It has a flatter trajectory than a .22 LR or 22 mag, but the 22 mag has more killing power due to a bullet with twice the lead.
I hope that helps. Consider that 22 mag ammo costs over twice that of 22 LR, and 17 HMR is more expensive than that usually. There are a lot more choices in bullets in 22 LR as well. My MK XXII loves CCI mini-mags. YMMV
Here is a picture of my XXII:
Please let us know what you decide.