First, check your hunting rules and regulations to see what calibers and actions are allowed.

I would use a .22 cal for the groundhogs as long as it is allowed since it sounds like there are many of them (cheaper ammo). if you are looking at close shots (under 125 to 150 yards) for the coyote, I like the .357 mag option with the proper ammo for downrange performance. Can you legally shoot groundhogs with .38 Special?