Unless you're driving, trains are generally the way to go. Summer journeys can be beautiful: the track is never far from a river or lake. From Petrozavodsk, the Murmansk line heads north via Kem, Kandalaksha and Apatity. From Vologda it's a straight 650km or so north to Arkhangelsk.
Buses provide inexpensive transport within limited areas but there are few long-distance services.
Intercity flights come and go. At the time of research the main regional routes were Murmansk-Arkhangelsk, Arkhangelsk-Kotlas, Arkhangelsk-Solovetsky Islands and (summer only) Murmansk-Solovetsky Islands and Arkhangelsk-Petrozavodsk.
If you're in a hurry and not pinching kopecks, hiring someone to drive you to a nearby city is a common alternative. You can usually negotiate a fair price, and the driver is bound to have some interesting stories to tell.
The Solovetsky Islands, Kizhi and Valaam are commonly reached by boat (from Kem, Petrozavodsk and Sortavala, respectively). These services are restricted to the ice-free summer months (usually late May/early June to some time in September or October).