Many variables to consider. Many states have different rules for minimum age for example in West Virginia there is no min age my son killed his first whitetail at age 10 my daughter at age 12.
A whitetail hunt might very well be the way to go, as they can be found in almost every state and prices for hunts will vary accordingly from a few hundred dollars to thousand depending on what you want.
Bears are an option in many states (guided and unguided, baited or non baited, hounds or no hounds depending on the state). Of course I'm talking black bears, looking at grizzly or browns is a whole different ball game. Pronghorn in the plains is always an option as well, plus the whole Mule deer, elk, moose options in some states as well. My suggestion would be do some research on what animal and part of the country you want to hunt in then let folks know what species/area you want to hunt and you will probably get better info that's more specific to where you are headed.