I truly think the 223 is the ultimate varmint caliber. It is widely available, proven, accurate, and deadly beyond several hundred yards.
On groundhogs, the 223 will open them up out to 300-400y if you know your stuff to shoot that far. (ie:wind, drop, ect) I just shot a crow the other day at 275y with 52g Speer hp's and what was left of it fell out of the tree. Cheap to shoot also makes it great for groundhogs as you can get a great amount of shooting per summer out of them.
On coyotes, the 223 is also a beast. A 40g V-Max at about 3650fps is ultimate for fur and will explode inside the animal, demolishing the internals. It will take a coyote out to 300 yards with the right person behind the trigger. This is pretty far though when you get down to it. Think about 3 times the range of your 100y shooting line.
The 204 is also a great cartridge and that will be my next varmint caliber. Slightly outranging the 223 and being flatter shooting I think this gun is better suited for reloaders as the ammo availibility is slim at the time. Also, there is a slight variety of ammo for it. All of which are for varmints only...nothing for 'no meat damage'. In my opinion, a very good prairie dog cartridge. Small frangible bullets moving fast as hell.... :twisted:
If one needs more range than a 223 will offer on a yote or groundhog, I think the "beyond 22 cal" cartridges are best for this. I have my 25-06 ready to kill a hog/yote as far as I can see one.