Georgia's Kakheti region is famous for its long tradition of wine making. You can buy good and cheap (€1-2) wine in recycled plastic bottles virtually everywhere here. There's also a fair chance that you will be offered substantial amounts of wine here at some point, but remember that trying to hitchhike while intoxicated is not very sensible!
East towards Sighnaghi, Azerbaijan
From the the bus/minibus stand, walk past it downhill and take the main road to your right. Follow this road for 15-20 minutes to an intersection where you turn left, walking downhill again. After passing some petrol stations this road passes through a forested, semi-urban area. Keep walking straight ahead until you see a big petrol station/service area on the left side of the road. Pass it for 25 meters and hitchhike on the right side.
This is quite a walk from the city center, so alternatively, hop on any marshrutka (minibus) heading east and just tell the driver to stop when you approach the petrol station.