My first post here then about Lithuanian scene heh
Mentioning Lithuania you probably mean either the capital city Vilnius or port of Klaipeda)
I can just give general advice, haven't been there for ages. Not much known about scene here compared to Chezh or Hungary for example, even Latvia/Riga is well known for. Saying that - Lithuania could be new exciting place to explore, not spoiled emerging market
Worth doing a research on this
General advice:
I am not aware of any "districts", but again, haven't been there for ages
Vilnius - must visit old town/city center Senamiestis (very beautiful and safe) and Radisson Blu Lietuva hotel (talk to concierge

then head towards south bank over the bridge, avoid north side Snipiskes and generally anything up north). Also avoid airport, train station and going deep into suburban areas
Klaipeda - cargo port (could be dodgy) and city center
hope this helps
