For only $200 your looking at used rifles. You might run across anything. Rifles are great for deer but your shotgun can also take a deer with buckshot or slugs. Pheasant are harvested with the shotgun using bird shot.
i would buy a .223, .243, or 270 but i dont think they are around that price range. you dont hunt pheasants with a rifle, but you do with deer. a 12 gauge pump is a great type of shotgun and will kill pretty much anything you want it to kill. it will kill birds,small game like squirrels, larger game like raccoons, big game like coyote, and even deer with buckshot or slugs. it would be a great gun for self defense, but also for hunting. you should check out your local pawn shop and talk to the owner about the guns he has and which one fits your need. good luck picking out a gun and good luck hunting!!!