The best way to move around Venice's waterways is by Vaporetto, and the closest stops to the Doge's Palace are San Marco and San Marco - San Zaccaria E and F.
If you're staying in Venice, the Doge's Palace is easy to reach by walking. The city is compact and pedestrian-only, so most central areas are within 15 - 30 minutes of St. Mark's Square. Walking also lets you take in the canals and narrow streets along the way.
Bicycles aren't allowed in Venice's historic center, but you can use them in the Lido or mainland areas. From there, it's possible to reach a vaporetto stop or parking area and continue on foot into the center.
