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, Vervet, and Fint in Tromsø and Bryggejentene in Ersfjordbotn.
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 and other furry activites at Tromsø Villmarkssenter: https://www.villmarkssenter.no/booking/ (select "summer activities" and if you plan to book shoot us a message and we'll try to fix a discount)
Sea kayaking and SUP: the best place to go is Sommarøy. Several companies offer both guided tours and kayak hire, for example https://www.sommaroyadventure.no/en/utleie or https://69nord.com/en/guided-activities-and-rental-of-outdoor-equipment-at-sommaroy/
Cykling: Tromsø Outdoor can set you up with bikes etc. - for a cheaper option check www.Finn.no to hire from locals.