Bratislava is the capital of Slovakia.
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East towards Slovakia
When going east you can take any bus because there is all the traffic going east. Still, people do not tend to pick up people standing on the side of the road. So practice your Slovak and ask at the service station.
South towards Budapest
When heading for Budapest you will see that the border is around 15 km from Bratislava. It is walkable (if you like that kind of thing) follow the motorway but be aware that once outside the city there are no petrol stations and only one parking place which is more or less between the border and Bratislava. The parking is far from ideal, very quiet, no services and so large its difficult to talk to drivers.
Not exactly hitching, but you can get a bus ticket to Budapest for 90 SKK (~4.50 USD) from the Student Agency on Obchodna Ulica. The place is, if facing towards the old town, on the left hand side of the street just before Postova Ulica, near the McD's/KFC and Slovak Pub. If you're under 26 don't forget to ask for the 10% discount. ����������������������������������������