Kaliningrad Oblast is now an exclave of Russia but it is populated by Russians only after it was conquered by the Soviet Union in the year 1945. Prior to that, the area was ruled by Germans. Ethnic Lithuanians made a significant portion of inhabitants. The area was part of Lithuania Minor and it is in Kaliningrad (then known as Koenigsberg in German and Karaliaučius in Lithuanian) where the first Lithuanian language books were printed.
History, however, is nearly swept away in the Oblast. All the placenames were changed. Even many river names were changed from German and Lithuanian ones to Russian (a rare practice after conquests). Instead of Bareiškiemis there is now Pervomaiskoe (literally "May 1st town"), in place of Koenigsberg - Kaliningrad (named after a famous communist Kalinin who never visited the city), Tilsit/Tilžė is now Sovetsk (after the Soviet Union).
In Kaliningrad, little remains of Koenigsberg as entire downtown was obliterated and only the long-abandoned cathedral is now restored by German donations. The massive medieval castle has been torn down and its place is now occupied by the Soviet Palace (actually a dull white tower block).
Some other towns were luckier, however, it is well understandable why German exiles used to weep when they saw their former hometowns after finally being allowed to visit them once the Soviet Union collapsed. Even those stately German buildings and churches that were not obliterated now lay in ruins. Not a single old town is intact. The few remaining old buildings dissonate heavily with the modern poverty surrounding them. They are joined by countless plain Soviet ones and ones rebuilt after 1945.
Chenekhovskoe (Insterburg / Įsrūtis), Sovetsk (Tilsit / Tilžė), Zelenogradsk (Cranz / Krantas) are among the more interesting towns.
In the times of Soviet Union travel to the heavily militarized Oblast was forbidden. Now this ban on foreigners still applies only to the town of Baltijsk (Pillau / Piliava). Still however, a Russian visa is needed and the waiting times on the borders may be long. They tend to be shorter in Neringa (Curonian Spit), but there you will have to pay a local tax for entering the national park on both sides. It may be quicker to get in by train from Vilnius. Oblast being an exclave every train from Moscow to Kaliningrad stops at Vilnius station.
Kaliningrad city itself may be reached by bus or train from Vilnius, and by bus from other main cities. Still, if you want to explore smaller and more interesting locations it will be better to drive a car.