I'd say first to give deer a try, it can basically be cooked like most other red meats. Other options are the game birds such as pheasant, dove and quail. It'll be more difficult to find locally, but Alligator is good too, cook it like you'd cook chicken.
As for how to cook it, venison burgers are great (mix in some bacon or other porkfat, deer is extremely lean). Other parts can be cooked like traditional steaks or tenderloin, sous vide is probably a good bet there for food safety and not overcooking it. For Alligator, try it deep fried, in my (limited) experience it doesn't cook as steaks very well. For the game birds, they're small and rather delicate, sous vide might be a good place to start too.