Greenville County is one large school district with many, many individual schools.
Many of the elementary schools are great, but as you move up, fewer of the high schools fall into that same category. In my opinion, you are better off finding the best high schools and then backtracking to make sure the middle and elementary schools that feed them are good.
Here is a link to the individual school report cards for Greenville County:
Greenville 01 - EAA School Report Cards 2013
You can use those to do some research on schools to get an idea of which ones have done well over time. You'll want to combine that with talking to people and looking at neighborhoods to narrow down neighborhoods you are interested in looking at.
Some of the high schools in Greenville County that have done the best over time include: Riverside, Eastside, Wade Hampton, Blue Ridge, and Mauldin.
Simpsonville has grown a lot over the last few years. In the past, kids around that area all went to Hillcrest but they now bus them to Woodmont and Southside as well so you need to really carefully check which schools a house is zoned for and then know that it could change over time. The re-zoning issue is one of several factors that led us to pick the Anderson District One area over Greenville County.