The type of game available for you will dramatically depend on your state’s environment and regulations. (Be sure you always have a license before you hunt!)
However, there are some general guidelines:
In the Eastern and Midwestern states, the most popular animals to hunt are white-tailed deer, small game (squirrel, rabbit, and more), upland birds (such as pheasant and quail), turkey, waterfowl, and black bears. Recently, elk populations have been established in some states (West Virginia, Pennsylvania, Kentucky, and Arkansas), and hunters can enter a lottery to hunt them.
In the Western U.S., there are more big game hunting opportunities. Common hunts include elk, moose, mule deer, or white-tailed deer. While most areas require big game hunting tags be awarded by a lottery or draw system, hunters still have a good chance. Some states even have over-the-counter hunting tags for public access (though generally only for archery equipment). However, small game, turkey, upland birds, predators, and waterfowl are also available for hunting.
As wild boar populations have grown prolifically, they have become more popular hunting targets, particularly in Southwestern states. Some states consider them to be pest animals, removing the limit on the number of animals that can be taken (unlike other big game animals).
Check with your state wildlife department to find out the particulars for your area.