Most one-day visitors don't use any transport: all of the major Old Town sites are within a reasonable walking time.
Orientation
The basic map of the city costs only 1 Euro at the TI. There are also maps available online (http://www.innsbruck.info...).
By car
Big parts of the downtown area are declared fee-based short-term parking zones. For longer visits, it is highly recommended to park off-site and use public transportation.
On foot
The classic walk into old Innsbruck follows.
From the main station Hauptbahnhof to the city center is a relatively short and enjoyable 10 to 15 minute walk. Walk out of the Hauptbahnhof, cross the street at the train station cross walk, turn to your right, and go down to the next street to your left. Walk on this street until Maria-Theresien Strasse, then turn right toward the city center. Taking this street all the way leads to the pedestrian zone and the Golden Roof.