From what I have seen, if you are looking for suburbs, Mauldin and Simpsonville are both nice and south of Greenville. If you want to be closer to the mountains, Traveler's Rest, Greer, and Taylors are nice. My husband and I moved here recently and love it so far
I would definitely recommend renting first though, just to make sure you like the area, before you buy anything at all though. It would stink to buy something and end up hating the neighborhood in 6 months.