The most easy way to get to Trakai is by using the car. The distance from Kaliningrad to Trakai castle is around 300 km in one direction. You will have to pay 20 new rubles for crossing the boundary from car (insurance) and by 5 rubles from each person crossing the boundary.  When crossing the boundary you should have the passport with yourself.

Road Map
Road map (click on it to increase the size)

If you want to stay for a couple of days you can do it in camping or in the hotel. Renting room in a hotel will cost you 60$ per night or you can rent a room in a state hotel only for 5$ per night.
It is pretty cheap to have a meal in a restaurant. And the castle is opened for tourist up to 19 o'clock everyday.

 
 Geographic note: Trakai — town on south-east Lithuania in 29 km from Vilnius. It is a regional center with the population of 6000 and with the territory area around 1150 hectares (550 hectares are the summarized lake surface area). The area of 4502 hectares across Trakai is protected by Lithuanian Government Law, cause it is considered as a  preserve area.

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