We actually did this route in reverse. It's easy to travel to Verona from Varenna en route to Venice.
We stayed in Verona for 3 nights & enjoyed walking around piazza Bra & Piazza delle Erbe. There is an arena much smaller than the one in Rome but still impressive. You can also visit the Duomo, Lamberti Tower, the Castelvecchio & it's museum, and of course,
the famous bronze statue of Juliet.
There is also the best gelato at Pretto (gelateria) on Piazza delle Erbe: http://www.gelatopretto.it/hd/
You will take a 'Regionale' train from Varenna to Milano Centrale where you will transfer to a Freccia (high speed) train to Verona.
Once you visit Verona, you will take a 'Regionale Veloce' train (8,60€pp) to Venice & you will get there in 1.5 hours.
You can check train schedules here:http://www.trenitalia.com
Type all train stations in Italian when using above website.
- Milan = Milano Centrale
- Verona = Verona Porta Nuova
- Venice = Venezia S. Lucia
Enjoy your trip!
EDIT: Depending on how many days you will be in Venice, you may want to take a day trip to Padua (Padova) & visit the Scrovegni Chapel (need advance reservations) & St. Anthony's Basilica. There is an open air market on Saturdays on Prato della Valle nearby the basilica. Padua is an easy 30 minute train ride from Venice.