By plane
As noted above, Murmansk and Arkhangelsk have domestic airports with services between them and Saint Petersburg. In Arkhangelsk's Talagi airport lands a airplanes from Luleå Sweden, Rovaniemi Finland, Tromsø, Bergen, Kristiansund Norway, Homel Belarus and Antalya Turkey. Air travel works in a pinch, but the fares are high while train fare is often very low, and you will likely have more inconveniences dealing with airport staff and what have you, than if you take the more standard overnight trains. If you are trying to get to Naryan-Mar in Nenetsia good luck, air travel is the only plausible means of success, with occasional service from Arkhangelsk.
By train
The principal mode of transport in Northwestern Russia and Russia generally is rail. Saint Petersburg is the undisputed rail hub of the region, and virtually all major cities in the region have rail service direct to the "Northern Capital." Southern cities also have direct service to Moscow. Given the vast distances throughout this sparsely populated section of the world, overnight trains are the most frequent mode of transport.
For intra-oblast travel, the most efficient and simple mode of transport are the electric train lines (Russian: ÑлекÑÑиÑки elektrichki. These small suburban radiate out from the main cities not just to suburbs, but to far-off smaller cities as well as small villages.
By bus
If the elektrichki don't take you all the way to your destination, you are in for an adventure. Buses and their smaller cousins Russian: маÑÑÑÑÑки marshrutki connect the rest of the small towns and villages to the broader transit system. It is rare, however, to see a bus that has a clearly marked route. Often the only way to know which bus to take, and for that matter when to get off, is to ask. If you don't have enough command of Russian to do so, consider finding a tour operator to arrange transport to your destination instead.