Klaipėda is a city in Lithuania on the coast of the Baltic sea and a major port in the Baltics.
Take bus 12, 7, 5, or 2 and then walk 2 km until the beginning of the road A1/E85 at the Jaku ring.
2015ish update: Since overpass has been built, Jaku ring receives mostly local traffic, if you try to catch traffic from overpass - it's dangerous due to high speed, merging lanes, noise walls and just not a happy place to hang out. 2.2km from the ring there are two gas stations, hiking to them should be ok but better get dropped off there initially.
Take bus 3, 9, or 6 to the final stop Miesto Ligonine (Hospital of Town). After getting off the bus, hitchhike from the bus stop across the street after the Lukoil petrol station.
West to Nida, Kaliningrad
It is easy to hitch a ride to the Curonian Spit from the ferry port for cars. Ferries leave about one an hour and cars have to wait in a queue. A ticket for a passenger or pedestrian costs about EUR 0.80. At night or when there is no security you might be able to sneak in without paying. If you want to go to Kaliningrad, ask cars with Russian number plates - there are usually a lot of them and they are likely to go there. You can check the schedule here.