Greenville has a very active homeschooling community.
As far as a classical school, there is the Greenville Classical Academy. They follow the university-model for upper levels, meaning your child goes to class 2-3 days a week, and does work at home the rest of the time. They also offer ala-carte classes for homeschoolers, though is all for upper levels. The elementary levels do the regular Mon-Fri schedule.
There's many co-ops too, if that is of any interest to you. Upstate Homeschool Co-op is a big one. HOPE co-op is another one that is pretty popular. There are other smaller ones as well. And of course, Classical Conversations co-op are aplenty round here. One on every block, it seems .
There's also a few "tutorial" type campuses...meaning they are not co-ops in the sense that parents do not have to participate..meaning it's more of a drop-off situation. Artios Academy, and Vanguard Homeschool Academy come to mind.
Rock climbing classes, First Tee golf classes, Lego building classes, horse riding classes, Homeschool P.E., Band, Dance classes, Archery team, Rollerskating Days, Ice Skating days, Sports teams, Choir, Art classes...etc etc...all of these and more are offered in classes that are just for homeschoolers.
There are many support groups that are just for "fun" and gathering together, that do not require co-op style participation. FCHome is a group on Yahoo that is helpful in terms of getting information...though they don't gather together often. Upstate Field Trip Friends is a Facebook group that is VERY active...field trips several times a month (you pick and choose the ones you want to go on), park days etc. The sister group to that is the Upstate Playdates (for Homeschoolers), again on Facebook.
We also have TWO homeschool-oriented book stores in the area. Lifelong Learning Resources is a homeschool store in the Lyman area. Learning Cycle is a new store, that is based on consignment...so used curriculum.
Let me know if you have any more questions.