Nantucket is a fairly large island by bike. You'll most likely just see the downtown area (cobble stone streets - I'd walk rather than bike), the ubiquitous lighthouse at Brant Point (don't forget to throw a penny from the ferry as you go by on the way home) and maybe out to the Cliffs. There are several bike paths of varying length. I'd stop at one of the dozens of bike shops at the ferry dock and grab a map.
The high speed ferry out of Hyannis is your best bet at $50 something round trip. Only takes an hour.
For lunch I'd try the Rose and Crown Pub if it's still there.