The Lake Erie Islands — Put-in-Bay on South Bass Island and the quieter Kelleys Island — are the obvious day trips, both reachable by ferry from the downtown waterfront or from nearby Marblehead.
Marblehead Peninsula, 20 minutes east, has the photogenic Marblehead Lighthouse (Ohio's oldest, in continuous operation since 1822) and a string of small lake towns.
Cleveland, 60 minutes east, makes a longer day trip — the Rock and Roll Hall of Fame, the Cleveland Museum of Art (free, world-class), and the West Side Market are all worth the drive.
Milan, 15 minutes south, is the birthplace of Thomas Edison and has a small Edison Birthplace Museum on the original site.