We will have several extra days in Venice. Any suggestions as to places to visit locally or maybe a day trip or two?
Padua, Vicenza and Ravenna are an easy train ride from Venice.
Verona is about an hour away by train. It's a beautiful city along the Adige River. Check out Rick Steve's Italy and read the Verona chapter. There's a lot to see there, and can be easily done in a day. The highlight would likely be the almost-2,000 year old Arena of Verona which is still used today for concerts and operas! Verona is also home to Shakespeare's Romeo and Juliet, has a portion of an ancient Roman road, the impressive Castelvecchio which can be toured and houses a lot of art and statues (and is beautifully lit at night), was home to Dante for a time, and has many other lovely attributes.
Lake Garda is about 20 minutes away from Verona and has some nice lakeside towns to visit (Peschiera, Bardolino, etc).
Padua is just 20 minutes from Venice and is the home to one of the oldest universities in Europe built in 1222. The likes of Galileo and Copernicus lectured there! Wow! Also has Basilica di San Antonio and the Scrovegni Chapel featuring frescoes by Giotto. Read up on all that in RS Italy also.
So much to see, so little time! Have fun!