I'd pick a center fire round for general purpose varmiting and predator control as well. I'd keep a good rimfire for those places where hunting with a centerfire would make hunting impossible or impractical. My thinking is it's better to hunt with a rimfire than not to hunt at all.
If I were just starting I think a .223 would be a good fit for all but the really long shots. Around here where I live, the range of the .223 is more than adaquate. It's about the cheapest round to shoot if you don't reload, and factory ammo is every where.
The debate on .22mag versus .17HMR has been around from the announcement of the .17HMR and you'll find many opinions. Personally, I've never had a .22mag that could come close to the accuracy of my .17HMR. That's not to say that there hasn't been some accurate .22mags or that the accuracy of guns chambered in .22mag might have improved in the last few years, but I've had several rifles in .22mag and was never impressed by there accuracy.
The accuracy of the .17HMR is impressive. You will find a lot of post by people who have had sucess killing coyotes with the .17HMR and others who have had miserable failures. I don't consider the .17HMR as a coyote round, but if one came by while I was out hunting with the .17HMR I'd sure find out for myself. I consider both the .22mag and .17HMR lite for coyotes. As both are lite then bullet placement becomes more important and that's where I believe the .17HMR to have the edge. I'm sure others will post with a different opinion on the .22 mag and .17HMR as their experience has lead them to believe differently. This discussion is certain to continue but has already been had, over and over.