A .243,30-30 or 7mm-08 would be good choices for Big Game/Deer...the Small Game would take a different Rifle as several mentioned so as not to damage it a lot.
A .22 Magnum would take out Squirrels and be big enough to take down a Coyote,the .22 Hornet would be a good choice also and would have a little more power and reach out further.A .204 or .223 would be a step up but would require head-shots on Rabbits or Squirrels but would definetly be better for the larger Predators like Coyotes,Bobcats and Foxes!
Lots of Hunters like a good Shotgun in a 20 Gauge or 12 Gauge for Rabbits and Squirrels but will limit You to closer shots if You want to hunt Predators!