A good starting gun for what you said would be a 243 if coyots is the biggest thing you will be after. But for snake or rabbit a .22 would be fine. Maybe go with a 22-250. That's a gun really for your second or third. The range on game you gave is pretty large though. Do you plan on eating the rabbits or just " Blowing them away " ??? If so the 243 would be fine.
Good Luck enjoy your gun