I have quite a bit of experience with show falconry (raptor husbandry for putting on shows and demonstrations of falconry without encouraging prey drive), as well as some hunting falconry. While hunting with falcons is certainly fun, I'd recommend a Harris hawk for more wooded areas. Females are large enough to handle the average hare, and their relatively social nature makes training them a bit less complicated (when raising a chick, you either imprint them to yourself or not. With Harris hawks, imprinting doesn't have as many drawbacks as it does with other species). Goshawks are also good for woods, but they're a bit more.. fragile.
Peregrines, sakers, girs, Lanner etc are all great falcon species to work with, and which one you choose depends on the hunting mode you prefer, as well as what prey you want to go for (peregrines will pretty much only get you birds). Training falcons is a bit tougher, and with these birds I personally prefer ones who aren't imprinted, as imprinted falcons are rather loud and very protective over their prey.
Golden eagles are awesome if you like hunting foxes and larger prey in general. Very fun- I know a guy that gets his eagle to fly alongside his Jeep for training.
