By plane
Murmansk Airport (http://www.airport-murmansk.ru) IATA: MMK has multiple daily flights to Moscow and Saint Petersburg, and limited services to other Russian cities including Arkhangelsk, Sochi and Rostov-on-the-Don. There are also several flights per week from Tromsø and Kirkenes. Seasonal flights are avalible from Helsinki too.
The airport is located about 40km south of Murmansk, near the town of Murmashi. Taxis to the city center cost R500-600 and make the trip in about 40 minutes. Catching a taxi waiting outside the airport is more expensive, expect to pay up to R1000. For cheaper and official taxiservice, you have to order a taxi, expect to wait up to 30-40 minutes for it to arrive, though. Bus 106 goes to the train station, stopping at Detsky Mir near the Poliarnie Zori Hotel on it's way, is less expensive but much more sluggish than a taxi.
By train
Murmansk can be reached from most places in north-west Russia by train. Moscow is 35-40 hours away and Saint Petersburg 27-30 hours, depending on the train. The Arktika ÐÑкÑика branded train is the fastest option, it also offers first-class wagons and a restaurant onboard. All trains make stop-overs in cities such as Kandalaksha and Petrozavodsk on their way. Other night trains reach Murmansk from cities as far east as Arkhangelsk or from Minsk and Brest in the west. There is also a local night train from Nikel close to the Norwegian border. Trains from Saint Petersburg and Moscow are daily, most others 2-3 times a week. During summer additional routes are added, mostly to Ukraine and the Black Sea.
The central train is located in the city center, one block downhill from Five Corners Square on ul. Kominterna, 16.