Piazza Grande (“The Big Square”) is the centre of this town, you can begin from here your tour of Gubbio. The principal monuments are Palazzo dei Consoli , Palazzo del Capitolo, the Cathedral and the Church of Santa Maria dei Servi… they will tell you the history and the legends of Gubbio.
Some worthwhile suggestions:
Visit the Basilica of Sant'Ubaldo, restored in 1500 (either drive or get the cable car, it's quite a walk up the hill).
Visit (walk to) the Roman ruins dating to the 1st century AD, or in July or August, go see an outdoor performance there.
Visit the archaeological museum housed in the Palazzo dei Consoli (see photo below). The museum is famous for housing the Iguvine tables (in Italian "Tavole Eugubine"), seven bronze tablets dating back to the 3rd century BC, which are written in ancient Umbrian.
Peek into Gubbio's Cathedral or a few of the other gorgeous churches around town.
