Easiest thing would be to go to the Sumter National Forest. Prospecting is allowed there, panning only though. There are several historic mines in the area, and placer gold in some creeks. You just need to request a free permit from the Long Cane Ranger district. They will give you a map with areas where you can prospect.
Outside of that, it's a matter of asking permission or maybe dredging in rivers. Most creeks will be considered private property. Mineral rights are usually owned by the property owner, but in some cases are not.
But regardless, I doubt you would find much gold in Summerville. Gold is found in SC above the fall line. Generally speaking, upstate of the sandhills is gold country. The Carolina Slate Belt and the King's Mountain Slate Belt run diagonally through the upstate, parallel to the coastline (more or less). Haile, Brewer, Barite Hill, and several other mines have uncovered major strikes, and much is still there. There is a lot of gold in SC, but it's mostly very fine or microscopic. I have read about some coarser gold and nuggets being found in the Western part of the upstate.
