Milledgeville will remain the Georgian capital for a while, at least. I could see Georgia being one of the big wheels in the CS government and the root of at least one of at least four possible regionalist blocs that would exist with any political parties, given how many of the CSA's founders were Georgians, and if the CSA winds up managing to create a lasting civil government Georgia would be one geographic stronghold of the super-states' rights crowd. As far as Georgia's development as a state...Savannah would be one of the big Atlantic ports of an independent Confederacy and something of a rival to Charleston in this regard. Atlanta would remain a major rail hub and probably expand in this regard in any independent CSA.
Georgia would have issues with at least three big cultural divisions among Georgian whites, with Georgian slaves being a fourth spoke of the wheel and one that is not necessarily fond of or getting along well with any of the white factions. There would be the urban-commercial set in Atlanta and Savannah which would naturally favor policies that favor the big cities. There would be rural divisions between the more rugged highland north with its smaller-scale pockets of Unionism (which would be a bit of a potential issue in inter-state politics), and the more flatland southern edge of the state, in all classes. There would also be the planter aristocracy used to dominating the state which would have even in 1862 an established tradition of butting heads with Richmond.