I've got a .223 and love it. .223 ammo is about the cheapest of any centerfire rifle, and it is the easiest to find. It also has the broadest choice of ammo selection. It shoots relatively flat out past 300 yards. The 22-250 and the 220 Swift shoot the same diameter bullet, just using a little more powder. They shoot faster and flatter, but can do a little more damage to the hides. 22-250 ammo can be found almost anywhere, but ammo for a 220 Swift can be hard to come by. Any of these rounds is deadly on varmints the size of coyotes or less. The .17 HMR is basically a .22 WMR necked down to a .17 caliber bullet. It won't have the range or the killing power of the larger centerfires. On small critters, its probably plenty. On coyotes, a .17 caliber bullet just isn't enough.

If you plan on shooting anything larger, like maybe deer, with the same rifle, you might consider a .243 or .25-06. These cartridges shoot very flat also, they just do it with more bullet. Relatively small bullets can be used for varmints, and larger ones can be used for medium sized game. The main downside is that these cartridges will damage the hides more than a .22 caliber centerfire. They also have a little more recoil (although it is still very manageable).