The old town is small enough to cover entirely on foot. The main pedestrian zone and the uphill path to Schloss Gayenhofen take under an hour combined.
For the surrounding valleys, regional buses depart from outside Bludenz Bahnhof into the Brandnertal, Klostertal, Montafon, Großes Walsertal, and Walgau. Timetables are coordinated with train arrivals. The Vorarlberg Verkehrsverbund (VVV) pass covers all these buses and is good value for a multi-day stay.
A car becomes useful if you want flexibility in the valleys — some trailheads and villages are poorly served by buses in the shoulder seasons — but is not essential for a visit focused on Bludenz itself.