Zurich packs an impressive amount of culture into a compact old town. The twin-towered Grossmünster and the Fraumünster with its Chagall stained glass are the two set-piece churches; the Kunsthaus is the city's flagship museum and holds one of the world's finest collections of modern European art; the Lindenhof hill gives the best view over the old town and the Limmat. James Joyce lived here during both world wars and is buried at the Fluntern Cemetery; a small but serious Joyce Foundation keeps the literary connection alive.
For something lighter, walk Bahnhofstrasse from the main station to the lake, poke around the guild houses along the Limmatquai, or take a boat out onto Lake Zürich. The Rietberg Museum, set in a villa in a lakeside park, holds an excellent collection of non-European art that most visitors miss.
Beyond the old town, the Swiss National Museum by the Hauptbahnhof gives the historical context for everything; the FIFA World Football Museum, the Museum für Gestaltung in Zurich-West, and the Lindt Home of Chocolate in Kilchberg are all worth the short journeys to reach them. For something quieter, the Sukkulenten-Sammlung on the lakeshore and the eccentric Villa Patumbah are both free and largely missed by tour groups. The Uetliberg mountain, 25 minutes by train, gives the panoramic view of city, lake and Alps.