Welcome to the forum pbala. For wildlife zoom lenses, there is just no limits to what someone can spend to get closer and sharper. Usually, the longer and brighter lenses are, the better. And the price goes with it. Also, fixed lenses usually are sharper than zooms.
With your 3100, remember that you' need "AF-S" lenses (lenses with an inside focusing motor). Third party lenses can be more affordable but you should start with how much you have to spend BEFORE starting shopping.
Good luck and share your pics!