Hveragerði is a small town and most of it is walkable from the centre in under fifteen minutes. The Reykjadalur trailhead is at the edge of town — you can walk there from the main street in about ten minutes, or there is a small car park at the trailhead itself.
A car is useful if you plan to use Hveragerði as a base for the Golden Circle or the South Coast — the distances involved make public transport options slow. Within the town, there is no need for anything beyond your own feet.