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.