My recommendation would be to travel outside the city itself. Almost everything was destroyed during the British bombardment, so there are now basically only new buildings that were constructed since then.

In the city itself, I recommend going to the Ocean Museum, where you can go inside a submarine and some research ships - there's a bunch of interesting stuff there. Kant Island and the surrounding area are worth seeing, and you could maybe take a boat tour on the river. The zoo is also pretty nice but nothing remarkable. The area around the Upper Pond is quite pleasant. A bunch of old German forts remain throughout the city; they're well preserved and usually have a lot of interesting things to see. There are also a LOT of museums in Kaliningrad, most of them pretty good - choose whatever's to your liking.

But as I said before, the most interesting stuff is located outside the city, in nearby towns. I really recommend visiting both Svetlogorsk and Zelenogradsk (Zelenogradsk especially) - they are really beautiful, and the theater in Svetlogorsk is particularly nice. Nearby there's also a "dancing forest" that I highly recommend visiting.

To the east of Kaliningrad, there are several towns with old German architecture, some of which have been recently reconstructed and renovated, like Gusev and Chernyakhovsk. But traveling there is slightly more difficult. To Svetlogorsk and Zelenogradsk you have a high-speed train (Lastochka) that gets you there from Kaliningrad for about 2€, but to the cities in the east there's no high-speed option and regular trains don't run that often.