Coyotes and woodchucks/rockchucks warrant one of the following:
.223 Rem, .22-250, .222, .243 Win, .204 Ruger, .22 Hornet or, if on a severe budget, a .17HMR which requires head shots on 'yotes.
My personal favorite is the .223 Remington, assuming this is a Varmint Only rifle. Loads are Available from 40gr all the way to 80gr bullets, with the best varmint bullets being from 45 to 68. This is a caliber that begs to be handloaded, there's enough accuracy in most factory .223 rifles that you can really wring it out with a tailored load.
If this is a Varmint/Whatever Else rifle, go with the .243 Winchester. Bullet weights are vailable from about 85gr to 120 gr. This takes you from varmints to deer with a change of ammo and scope zero. Honestly, though, I think it's a little much for just varmints.
The 22-250 is O.K., but usually more picky about ammo and at one time had a reputation for being hard on barrels.
.222 was replaced by the .223. Some benchrest shooters may argue that it's more accurate than the .223, but I've yet to see real proof.
.204 Ruger is a proprietary cartridge whizzing a little .20 caliber, 35-40gr bullet along. Absolute death ray on woodchucks. I can't recommend it only because I have no idea what it does for coyotes. If someone else can claim good results on them, put this little thing on your short list.
.22 Hornet. If this were thirty years ago, I'd say go buy one. Ammo is hard to find on the shelf right now, though. When you do, it's pretty spendy. If you want to handload this one, get a mess of cases, some 45gr bullets and a few cans of Hodgon "Li'l Gun" powder. If you handload, this is a contender. If not, you won't hardly be able to afford to shoot it.
.17HMR is a last resort, "I can't afford a centerifre so here's what I got" varmint rifle. I like it for squirrels and woodchucks under 200 yards, but wouldn't dream of using it on anything heavier without a guaranteed head shot. It's a .22 Rimfire Magnum necked down to .17 caliber. Notice I said "Rimfire". Everything else here is centerfire.
Now for rifles. Personally, I think you either have to go with the Remington 700 or Savage/Stevens rifles. Both have a reputation for excellent accuracy, both are push feed, both are available off the shelf in nearly any caliber you want, especially .223 Remington. Here's where the coin turns in favor of the Savage/Stevens rifle. The barrel has a much fater twist, 1:9 vs 1:12 for the Remington. This allows you to use the heavier bullets if you reload, the heavier loads off the shelf if you don't. Couple that with the fact that a Savage Model 10 with a scope is the same price as the Remington without, I opt for Savage. You can go even better, buy a Stevens Model 200 (same rifle as Savage Model 10, but plastic stock and no accu-trigger) for $300 and put better glass on it yourself.
WOW, all that to say "Go buy a Savage .223".