Howdy and welcome to the forums.

Yes, the .223 is an excellent choice for varmints. That's what I use. As for the gun, I'm not a big single shot fan, either. There are a lot of good, quality bolt guns out there that can be bought at pretty reasonable prices. And like DM1975 said, be prepared to invest in a good, high quality scope. If you wanted a rifle for deer/predators, you might consider something larger that still shoots fairly flat, like a 25-06. It is a little harder on the pelts, so if your keeping the hides, it may not be an option. I don't keep the pelts (mainly because most of the coyotes I shoot don't have much hair), so excessive damage is not a concern for me.