Gori is a larger town for Georgian standards, but 15 or 20 minutes of walking will get you out of the town center and to a decent spot to hitch to the highway going between Tbilisi and Kutaisi. This highway is about 3 or 4 kilometers outside of the town center.
One of the busiest highways in Georgia runs past Gori.
From the center of Gori, you will probably need to walk out toward the highway before you start hitching. Stalinis Gamziri (Avenue) runs from the center past the Stalin Museum. If you keep walking past the museum, keeping it to your left, you will be heading for the highway. The road curves a couple of times, but if you stay with the major flow of traffic you shouldn't have any problems finding your way. Once you hit the highway, if your ride isn't going your direction, there is a great hitching spot with lots of pullover space at a petrol station on the highway close to the turnoff.
When arriving in Gori, look for the old ruined castle that is on a hill, basically in the city centre. Really, you can't miss it, as you can see it from anywhere in the city. Walk up there and just ask the guards if it's ok for you to pitch your tent there. They speak a little bit of English and they will most probably allow you to sleep there.
Another good option can be found right next to the Cave Town of Uplistsikhe, 10 Kilometres east of Gori. If you are on the road leading directly to the Museum area you can find two small gorges in the mountains to your right. In the one closer to the entrance you can find a very nice spot hidden behind some trees and bushes. IF you dont want to walk or hitch you can take a cheap bus from Gori to the village next to the historical name (ask in the Tourist information).