Kaunas is a city in Lithuania.
To Vilnius[]
When going to Vilnius you take a bus to the highway. There is a bus stop on the highway where it's possible to get rides, but it might also be a good idea to not deal with the hitchhiking competition and place yourself at the entrance of the highway.
Quite far away from Vilnius there is a huge hypermarket next to the highway with a free bus to the centre.
To Poland, Latvia or Klaipeda[]
When going to Poland you take a bus to the end of the route. Make sure you make it to the Polish border. From there you will easily find a long ride.
Alternatively get a bus from the centre to IX Fortas (I think its bus 23 or 29, but I'm not sure, look on the maps at every bus stop), this is the main road where the motorway to Vilnius joins up with the E67 (Via Baltica north and south) - with a PL sign you should easily get a lift from a truck to the border. Note that this border is *very* slow for trucks to cross and there's always about a 2-hour delay, so when the truck stops, get out and walk (or try to hitch a normal car) to the border, just after the polish side there's plenty of room for anyone to stop.
Heading for Latvia this road also goes to Klaipeda, so if you just hitch signless the road that branches off to Klaipeda is about 20 km from here, so you're not sure where you'll end up. Note, that this spot is usually quite busy for hitching - last time I was here there was 16! people also hitching, so holding a sign to your destination will make things faster.