The closest city is called Konstanz (Constance) which is 5-10 minutes away from Mainau Island. Official site: www.konstanz.de
stadt.konstanz.de/tourismus/englisch/
City Map Constance
Travel/Continue by car:
Friedrichshafen -> Konstanz -> Mainau Island
• From Friedrichshafen towards Meersburg on the B31
• In Meersburg follow the signposts to the ferry
• Pleasant ferry trip (15 min.) to Constance
• In Constance follow the signposts to the Mainau Island
 
Freiburg -> Konstanz -> Mainau Island
• Approaching from Freiburg on the B31 towards Tuttlingen
• Take the A81 in Geislingen towards Constance
• At the motorway intersection follow the directions to Constance
• Follow signs to Constance on the B33
• Shortly before you reach Constance, follow the
  signposts to the Mainau Island
 
Stuttgart -> Konstanz -> Mainau Island
• Approaching from Stuttgart
• At the motorway intersection Stuttgart take the A81
  towards Singen
• Follow signs tot Constance on the B33
• Shortly before you reach Constance, follow the
  signposts to the Mainau Island
 
Zürich, Switzerland -> Konstanz -> Mainau Island
• From Zürich towards Kreuzlingen on the N1->N7
• After the border Kreuzlingen/Konstanz follow the
  signposts to the Mainau Island
Travel by plane:
closest airports are Friedrichshafen (FDH), 25 km and Zürich-Kloten (ZRH), 75 km, hourly train to Konstanz.
Examples for best connections:
• Portland (PDX)->Frankfurt (FRA)->Friedrichshafen(FDH), operated by Lufthansa, a United Airlines partner
• Seattle (SEA)->one stop within the US->Zurich (ZRH), operated by American Airlines, Delta, KLM, SAS etc. or
  SEA->PDX->FRA->FDH operated by UA/Lufthansa
• Monterey (MRY) ->LAX->Zürich (ZRH)