There are several possibilities to get to know the City. Follow your intuition, or use one of these offerings:
A Guided tour through the Old Town with the "Nachtwächter".
Using the yellow train gets you comfortably through the inner City and the Old Town.
A sporty version: Using a Segway to get around Linz. "LinzErSchweben"
Austria Guides, the professional Tourist Guides offer individual and special Tours around Old Town Linz.
Any way you will encounter lots of interesting spaces.