My personal favorites were the Tromsø Museum and the Tromsøbadet, the latter of which I visited twice.
There is also shopping available on the island for souvenirs and otherwise. I enjoyed the second-hand and vintage stores of UFF Second Hand Tromso and Edel Antikk og Vintage. To refuel, I sipped coffee at Tøllefsenhjørnet and picked up baked goods at Backstube.