Hello! I'm from Lithuania, and I suggest to visit of course capital Vilnius (there is Gediminas castle, beautiful old town, cathedral, churches). Near Vilnius you can find other amazing small town Trakai ( there are beautiful castle in the middle of the lake, where you can rent boat or water bikes. Also in this town live small ethic group of people karaims, so you can try their amazing food, like kibinai).
From Vilnius you can go easy by train to Trakai, Kaunas and even visit cities near the Baltic sea.
Kaunas is the second biggest city of Lithuania, there are a lot of museums, churches, also castle.There are Kauno marios (lagoon) where you can sail.
Near the Baltic sea you must visit Kurių nerija. It's amazing! this region is in UNESCO. There are beautiful dunes, the city is very quiet and peaceful.
Some other interesting places: In iauliai - Kryių kalnas, Druskininkai, Nida...
If you want action, you can visit water park in Vilnius called Vichy park. Also in Lithuania there are a lot of parks, where you can climb trees and fly with rope above the river. One of the biggest is Tarzanija park 
Traveling is easy by train, and it's cost less. From Vilnius you can go to some mentioned cities like Trakai, or travel through all Lithuania and see the Baltic sea. From Vilnius to Klaipėda (near the sea) you will pay about 15euros.