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 there is an interection with a sign for Samtredia.
From the city center, you can also walk the main road in direction road. There will be a huge building with a Cola-Cola sign on it. There you can just stand on the road side.
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 border, 1 ticket is around 80 Tetri.
The road south enters Turkey at the Sarp-Sarpi border crossing.
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.