Close to downtown, shopping, dining, and great schools... I'd suggest either the Augusta Rd/Alta Vista/Country Club area or the Parkins Mill area. Both are well established areas in the City of Greenville and within 5-15 minutes of downtown (with Augusta Rd area being closer). The Parkins Mill area is going to give you rather large lots and more room to breathe. I'd suggest begin looking at those two areas based on your price range and requests. The schools are good and the access to downtown and the shopping corridors are unbeatable.
Not to mention the beautiful shade trees that line the streets of these area.
For what it's worth, I'd definitely try to locate in the City of Greenville. Greenville has a very high quality of life that's unmatched elsewhere in the metropolitan area. There's a great network of parks and greenspace, a good network of sidewalks, and a strong network of bike lanes/paths. Sure, you can enjoy these things by driving to them, but living amongst these amenities is really worth it, IMO.