As a visitor I’ve found it easy to learn how to get around Kaliningrad.
Getting to the city from the airport is easy; if you don’t want to wait around for the bus then there are two taxi booths near the building exit where you can hire a taxi for a fixed price (500 rubles at time of writing). You pay the lady at the booth and get directed to a taxi and given a receipt which you give to the driver. Having your destination written in Cyrillic script is a good idea.
In the city your options are taxis, bus, tram, or mini bus (“marshrutka“).
You don’t flag down taxis but find them at a taxi rank, or call the company to order one. Though there might be a meter it is usual to give your destination and negotiate a price before you start off. Uber doesn’t appear to have arrived here yet but there is a smartphone app called rutaxi that is simple and easy to use if you have a Russian SIM card. Rutaxi tells you the price when you order the car (so no worries about being taken the long way round) and it is cheaper than a regular taxi. Pick up and destination points can be entered by using a Russian keyboard on your phone or simply (my preferred method) choosing a point on a map. You get a price quote and after accepting you get info on the car model, colour and license plate. I’ve found it easier being picked up in a quiet place as it’s easier for the driver to spot you then. Don’t rely on regular or other taxi drivers to have change, they never seem to have.
Routes for buses, the tram (now only one line sadly), and mini buses can be found at website. The site is in Russian but using Google Translate all becomes clear.
Another very useful app to have is 2gis app. Using 2gis with its Kaliningrad map you have a very detailed street map of the city but also a route finder with the possibility of giving you suggested public transport options.
Buses are fixed fare at 18 rubles and the fare is collected by a conductor, this applies to the tram as well. They stop at fixed points on the route. Bus stops themselves rarely have any indication of the numbers of the buses that stop there, you just have to know. The tram is slow; think of it more as a good way to travel though and see the city if you have the time available rather than a means of getting from point A to point B. Mini buses will stop to pick passengers up and let them off at any convenient point in areas outside the centre. In the centre itself this isn’t possible because of traffic conditions, and they tend to stop at the same places as the buses. The fare is 22 rubles and it is paid to the driver on exiting the mini bus.