Georgia (country) > Batumi
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From the city centre, walk northeast on Chavchavadze St. under you see a tall building with a Coca-Cola advertisement on top of it. Another characteristic point could be a catholic church on the right side of the road. You can thumb here, though you'll have to be careful not to attract the many marshrutkas and taxis driving by. This is after the bypass joins the main road, so you might even stop a truck.
East to Akhaltsikhe
From city centre go in direction Turkey and soon you'll come to the crossroad where you go to left following sign Akhaltsikhe. Road is going through mountains and it's really bad quality. After lets say first 60km there are even less cars. It can take you a long time to cross the mountains because there are only small villages and locals usualy travel by marshrutka. Even when you take a car it will be very slow ride because there is no asphalt on the road. From village Zarzma (30km from Akhaltsikhe) there is really nice asphalt road and also more cars to hitchhike. In Zarzma there is also a monastery and if you ask monks they might offer you a dinner, warm shower and bed for the night(for me it worked, but for sure it's good if you speak Russian or Georgian).
Note: not to be attempted in winter!
South towards Turkey
You can walk towards the south of Batumi and get on the big road to Turkey. Hitch a ride with anyone going to the border and cross it on foot. The wait won't be long if you're not from Georgia or Turkey. From there, go right and hitch along the coast.
Or take bus from the Central Bus Station, it goes directly to the Sarp-Sarpi border crossing
In Makhinjauri (you can get there with local buses, number 10 or 101) there is large beach (of little rocks, not perfect as sand ones but there's no sand in black sea..) away enough from the city and easy to reach. There are also minimarket to grab/buy meal and even an internet cafe. Also a good spot to start to hitch to day after.
It MAY BE possible to pitch your tent in front and to the left of the Hilton Hotel facing the sea. There are a lot of other tents so you can meet locals and other travelers and you will not be bothered by the police.
It also has been reported that camping around the alphabet tower is possible.