Generally, most things in Tromsø's compact centre are within walking distance. However, there is also a good network of buses. In summer, you can rent bikes, and in winter you can rent cross country skis, both allowing you to roam the built-up areas of Tromsø.
By taxi
The rest of the time, there are plenty of taxis. Call them at 77 60 30 00. It is, however, cheaper to just go to a taxi stand and pick one up. Taxis are metered, and completely safe.
By bus
Buses are plentiful and very reliable. You currently pay NOK 28 for a one hour ticket.
Note that many routes has the city centre in the middle of their route, therefore it is essential to catch a route in the right direction. F.ex. 42Stakkevollan is driving to a residential area on the Tromsø island, 42Storelvis driving to Kvaløya. The ride from Storelva to Stakkevollan takes 45 minutes.
From the city centre:
Bus 28Solligården, 26, 20 Kroken, and 24 Kroken are found in the Sjøgata/Havnegata street just down below the Torget Main square. Any one of these is good for the Arctic Cathedral Ishavskatedralen.
Bus 26 goes to the Cable Car from Peppe's Pizza near Torget The Main Square. Ask for a "Fjellheisbillett" Cable Car Ticket. This includes a return bus ticket and the Cable Car ride, and is cheaper than buying each ticket individually.
Bus 37 goes to The Tromsø Museum. It leaves from Fredrik Langes Gate, just down below the à hléns outlet.
Bus 20Stakkevollan and 21UiT/UNN goes to the University. For the Botanic Garden, take the 20/21 to the Bankrupt Planetarium, walk down the nice foot path, enjoy the Garden and take bus 42Storelv back into town.
Bus 34 from the southern end of Sjøgata opposite Dolly Dimple's, just up from the Tourist Information for a tour of the Island. It takes you around the southern tip to the shopping centre of Jekta, from where there are lots of buses back into town: 24 Kroken sør and 26 Kroken tae rather long detours through several residential areas 26 also passes the Carmelite nunnery, 28 Solligården and 40 Sentrum take a somewhat more direct route, while 42 Stakkevollan takes you to Polaria and then downtown through a tunnel with two roundabouts inside. Lots of scenery and cityscape for 26 kroner.
Bus 20Stakkevollan from Fredrik Langes gate or 42 Stakkevollan from Sjøgata to Stakkevollan Skole, walk up the hill at the water reservoire and watch Northern Lights to the north less light pollution than elsewhere on the island.
Bus 20Kroken or 24Kroken from Havnegata to Tromsø alpine centre