Central Murmansk is compact enough to walk in twenty minutes end to end. For longer distances, the city has an extensive network of trolleybuses and marshrutka minibuses — frequent, cheap, and warm, which matters when it is -25°C outside. Taxis booked through Yandex are reliable and inexpensive.
For the Alyosha Monument, expect a steep walk up from the trolleybus stop at the base of the hill. The icebreaker Lenin and Marine Station are an easy walk from the centre. To reach the Khibiny Mountains or the Sami villages around Lovozero, either rent a car (4WD recommended outside summer) or join a tour from one of the operators around Five Corners — public transport thins out quickly beyond the city.