Northern Italy in mid-March

Not sure you'll see a lot of "cold, rain, snow" in Bergamo in March.
Maybe rain, but in March you'll likely see the weather of San Francisco in February (about 40F at night and high 50s to low 60s F during the day). In almost 30 years around here I've yet to see snow in the streets of San Francisco.
If the weather is good, you are very close to Lake Como and Lake d'Iseo and other lakes are not far (Lake Garda, etc.). The Alps are not too far (although you might see snow at higher altitudes).
Bergamo is very beautiful, but several cities in the Northern Italy are nice to visit, in addition to Venice, which you've seen already: Verona, Milan, Parma, Mantova (Mantua) are all within 1 to 2 hr drive (or train). Not much farther afield you can see Vicenza, Padua, Ferrara, Ravenna, Bologna, Turin, etc.
Good food, wine, history is everywhere. You are in Italy, what do you expect to find, bad food with no wine and no history?

To give you maximum flexibility I would suggest 3 nights in each of Verona, Padua and Bologna.
Spend your final night back in Bergamo for ease of access to your departure flight.
Visit Milan as a day trip whilst you are spending your initial 5 days in Bergamo.

Would Lake towns like Como etc even be worth a visit in mid-march? Would anything be open?

thoughts on Piedmonte area? Just throwing it out there.

I think Turin is a brilliant city and sadly underrated by many here (most of whom have never visited). It has great food, a lovely city centre, a world class Eyptian museum, a stunning museum of cinema, a superb automobile museum, and beautiful churches and palazzos. It is also not overrun with tourists and along with Bologna is a great place to visit.

It also is the home of Juventus and Torino football clubs. Try and get to a game - the atmosphere is electric.

Take a look at the Colli Euganei near Vicenza, great wine and food. Or Bologna, if you haven't been yet, with day trips to Modena and Parma.

I found Bologna was an excellent base for day trips by train and I enjoyed the city itself very much. Don't skip Ravenna, a pretty town with fabulous 5th-6th century church mosaics. Verona has lots of interesting sights (don't miss the Arena) and is charming in the evenings. Padua has enough sights for a full day, though the town itself wasn't one of my faves. I spent a night in Ferrara - lovely old town, with hardly a car in sight, everyone bikes.

Check if there's anything special you want to see (or avoid) during the holidays - Palm Sunday through Easter Sunday (March 27).

Awesome. Thanks for all the tips!