If you expect to be serious about taking on coyotes, any rimfire is going to be too small unless you're calling them in and calling them in close. On the other hand, if the majority of your shooting will be targets and hunting small game and at short ranges, the .22 rimfire is ideal.

I own a Mark II target, as well as a variety of .223/5.56mm bolt and semi-autos. If you're going to "target shoot" with a centerfire, the .223 is hard to beat. HOWEVER - I don't often see many fundamentals reinforced with younger shooters when they show up with dad at the range with an AR and a pile of 30-round mags. My hearing protection gets a workout, but that's about it.

My recommendation would be a .223 bolt gun (Remington 700 SPS Varmint might be a good place to start looking at specs). Most manufacturers offer a .223, though. I would expect that you could find a new rifle, GOOD mounts (spend the money), and a decent, mid-level optic for about what you'd invest in a new AR with iron sights.

In the end, you'll have something perfectly capable of taking coyotes, as well as a platform from which to focus on the fundamentals at longer ranges, rather than a machine for turning propellant into noise.