Most common birds now days are Harris Hawks, Goshawks, Coopers Hawks Red tailed hawks, in the falcons Gyr Peregrine Hybrids, and both peregrine and prarrie Falcons
on small birds people fly sharp shinned hawks, merlins, Coopers hawks and kesterels, Kites and hawks like that mainly catch grass hoppers and bugs not very exciting....
In Idaho the only raptor that you can not use is a Ferruginous Hawk I do not know why but they do not want you to use them now other states may have different laws, a licened Falconer is allowed to take young birds from a nest then they are banded and raised as a imprint in other words they think they are human...
It is a very rewarding sport but some things to consider you will be a apprentice for two years flying only a red tailed hawk you will have to build a mews (a hawk house) to keep your bird in and buy about 500.00 bucks of equipment... in addition you must have a prey base to fly your bird on which with red tails could be squirrels,or rabbits and have the time and gas money to fly the bird at least 3 times a week.....