Almost everything you came for is in the old town, and the old town is mostly pedestrian. From the train station it is a fifteen-minute walk over the Staatsbrücke into the medieval lanes; the bus network (Obus, mostly trolleybuses) is dense and cheap, and a single ticket is good for an hour with transfers.
The Salzburg Card (€32 for 24 hours, €45 for 48) covers all public transport plus most major sights — the fortress funicular, the Mönchsberg lift, every museum on the cathedral square — and pays for itself by the second museum if you are doing any sightseeing at all. Taxis are reliable but pricier than in Vienna. Bicycles are an excellent option in summer; rental shops cluster around the Hauptbahnhof.