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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. 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.
Alternatively, if you don't mind taking a marshrutka and want to take the slower road through Ozurgeti to Samtredia, a good strategy for hitchhiking northeast is to take a marshrutka from the taxi stand at the eastern end of Chavchavadze St., where it joins the main coastal road, to the town of Kobuleti about 20 km north. Get off at the first (southernmost) stop in Kobuleti, follow the northeast road for a couple of hundred metres, and just after the railroad tracks you can hitch.
South towards Turkey
Just 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 101 from the Central Bus Station, it goes directly to the Sarp-Sarpi border crossing and a ticket costs around 80 tetri.
The cheapest sleeping is around the bazaar, where there are some cheap hotels and nearly every family rents out rooms. Even in high season a room for two people can be had for 20 lari. There are also some cheap hostels. 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