Mainz's Altstadt is the old heart of the city, with half-timbered houses, narrow lanes, wine taverns, and small squares around the cathedral. It is the best area for a first wander in Mainz, especially if you want the city to feel older and less formal than the shopping streets around it.
Look for the Augustinerkirche and the lanes around Kirschgarten, where the rebuilt old town still gives a good sense of Mainz's medieval street pattern.