As mentioned a .22LR Caliber would be tha way to go if Your wanting to Hunt Rabbits and Squirrels,they are easier on the meat and can carry a long distance,one thing to remember is where Your bullet will hit or end up past an Animal or Target,always be careful because a .22 Caliber can travel a very long distance some 1 to 2 miles away.
Another caliber to consider would be a .22 Magnum,it has more knock-down and will shoot further out and flatter and can be used for Coyotes also.A Savage bolt-action Rifle would be a good choice and reasonably priced,Marlin and Ruger are also good brand name Rifles.
For Hunting Predators like Coyotes,Foxes and Bobcats there are many Calibers out there to choose from,a lot depened on how far You plan on shooting and how much knock-down power a Gun/Rifle has and how much recoil too.
.17 HMR Rim-Fire (flat shooting but not much knock-down power for larger Predators)
.17 Remington Fireball Center-Fire (flat shooting and minimal knock-down power)
.22 Magnum Rim-Fire (will reach out a little with some better knock-down power)
.22 Hornet Center-Fire (flat shooting a little further,good for most Predators)
.204 Ruger Center-Fire (fast-flat shooting Caliber with good knock-down power and little damage to hides/pelts). light recoil
.218 Bee Center-Fire (flat shooting good for distances)
.222 C-F (long distances,good knock-down power)
.220 C-F (another good caliber)
.223 (good caliber with nice knock-down power)
.22-250 Center-Fire (flat-shooting good for long distances and great KD power)
.243 Center-Fire (good caliber with good KD power),little harder on hides/pelt tho.
I'm sure there are a few more Calibers I have forgotten or left out but these are all good Caliber Rifles that would work great for Coyote Hunting,depending on Your needs and Hunting situations,some will reach out further and flatter than others.
Great thing about different types of Hunting is You get to have an excuse to buy a New or Different caliber Rifle.