Deer have come back to Oklahoma in a big way. Size of the deer run probably a little bigger in weight than Texas whitetails. Oklahoma does not have a reputation of producing large racks, but some nice heads are not uncommon. Most all of the decent hunting is on private land, but if you work to cultivate a landowner relationship, it is still easy enough to find a place to go. There are archery, muzzleloader, and rifle seasons, and the number of days to hunt continues to increase as the number of deer increase. State Wildlife is attempting to control deer numbers by advocating more and more doe harvests.
Quail numbers vary a lot depending on areas of the state and habitat. West of OKC used to be some of the best quail hunting anywhere in the U.S.
Pheasant are becoming very common, especially in farming areas closer to the Kansas line.
Ducks and geese can also be very good, if you find the right place to go.
Prairie chickens used to be common, but have pretty well gone the way of the dodo bird, due to loss of native prairie grasslands.
Oklahoma is typically a very hunter-friendly state, but of course individual cases vary individually.
My advice would be to try to establish a landowner relationship in the off season, and I think you would be able to do well and have plenty of days to hunt.
Good luck,
Bob