Adventures

In Tromsø

Sauna and harbour dip: https://www.pust.io/badstue/tromso/

Northernmost brewery in the world + polar bear: https://www.mack.no/en/index.php?p=about

Sherpatreppa/fjellheisen and hike to Fløya (onwards to Tromsdalstinden/Hameperokken if you're feeling "frisk")

Best coffee shops: Risø, Tøllefsen, and Fint. 

Best museums: Polarmuseum, Tromsø University Museum, Planetariet (at the university).

Best seals: Bella at Polaria

Climbing

For trad and sport climbing in Tromsø check out: https://klatreforer.tromsoklatring.no/areas (select "Troms)

Indoor climbing wall (+ pool and spa) at Tromsø klatresenter

Indoor bouldering (family friendly) at Buldreterminalen

Hiking

There are lots of hikes all around Tromsø. Check ut.no for a good map and overview over Norway's hut network. Ut also has some hike itineraries, but it is far from comprehensive. 

Kugo.no has a really good overview of summer hikes ("toppture") with maps and route descriptions. The site is in Norwegian, but should work with google translate. You might want to double their time estimates unless you're running ;) 

Other outdoor activities

Husky hiking etc. at Tromsø Villmarkssenter: https://www.villmarkssenter.no/booking/ (select "summer activities")

Sea kayaking: the best place to go is Sommarøy. Several companies offer both guided tours and kayak hire.