Kaliningrad is the city that changed its mind about itself. It was Königsberg for almost seven centuries: a Teutonic Knights fortress founded in 1255, the coronation city of the Prussian kings, the town where Immanuel Kant was born, lived his whole life and died. In 1944 British bombers destroyed the old core, in April 1945 the Red Army took what was left, and in 1946 the city was renamed after a Soviet official, Mikhail Kalinin. The Germans were expelled, Soviet settlers moved in, and for decades foreigners were not allowed near it. Today it is a Russian city of about half a million that sits 663 km west of the rest of the country.
That history is the point. Three days is the right length because the sights are few and spread out: one cathedral, one lost castle, a handful of surviving gates, one very good amber museum, and then the coast. There is no old town to wander for a week. What there is, is a place where Prussian brick, Soviet concrete and recent reconstruction share a street, and you have to read the signs to know which is which.
The plan: day one is Kant Island and the river, day two the amber, the gates and the castle hill, day three a run up the coast to Svetlogorsk, Zelenogradsk and the Curonian Spit. Start on a Wednesday: the World Ocean Museum's ships and gates close Monday and Tuesday, and the Amber Museum closes on Mondays from October to April. Read the visa and access tips first, because getting in is the hard part. For the country as a whole see Russia.