The town is very small and eminently walkable. The distances from the town to adjacent towns is generally further than walking distance. Thus, you would want to study the bus system that can take you all over, bring a bike or have a car. There is a bike rental in town for the active ones that don't want to walk around.
The lone exception is Hnifsdalur, a tiny village of 250 people which is situated near hiking trails. Hnifsdalur is approximately 5 km away from Ísafjörður, and is an easy walk on a paved walking trail running adjacent to Road 61.