how many days to spend on Lake Como?

I've stayed in Varenna twice, each time for 3 nights. It depends on what you want to do there, but I found that to be an adequate time. I like Varenna for the town and ease of arriving a departing by train. I wouldn't bother with a day trip to Milan. I spent a day/night there two years ago before an early flight and wasn't impressed. But that's me. If you must go I would spend a night there on the way to Venice.

If you're looking for something fun to do while you're at Lake Como and you like to cook/eat, the cooking class at Il Caminetto is a lot of fun. Reservations are required and I'd suggest doing it plenty ahead of time. http://www.ilcaminettoonline.com/lezioni_e.html

3 nights would be good for me. I am sure some slow travelers would disagree but I didn't think of it as a place with that much to do. Relax and site see around to the little pretty towns along the lake.
If you plan on using up a day to see Milan, while staying in Lake Como I would change that to 4 nights.
Takes a couple of hours I believe from the towns you likely want to stay in on the lake.

I would take the ferry from wherever you are staying to Como the town/city and then take the train from Como to Milan ; think that would be the best option for going and return and the ferry is obviously quite scenic and enjoyable.

There is a lot to do on Lake Como. We spent two weeks on our second visit there and never lacked for places to explore. Take the ferry to the many villages to the north and south parts of the lake, many gardens and villas.

The train from Varenna to Milano Centrale takes just over an hour. I can't wait to visit Milan because so many people seem to hate it. How can The Last Supper and Michelangelo's Ronandini Pieta, and the Duomo Milano not be worth a one hour train ride?

Our favorite place after four trips to Lake Como is Villa Balbaniello and it’s gardens.

I was thinking the same thing How could the last supper not be worth the trip?

I agree with Rick that the appeal of the towns at mid-lake on Lake Como, is precisely their lack of "turnstile attractions." The point of going there is "seeing how low you can get your pulse," and "if relaxing's not on your agenda, the Lakes shouldn't be either." It also makes a very nice "vacation from your vacation." (all phrases in quotes are Rick's).

I spent 2 or 3 nights there (I'm not at home where my records are), and it was lovely. I stayed near Varenna and saw Bellagio, but didn't get to Mennagio (next time).

I like Milan a lot. It seems to upset people because it doesn't conform to their idea of "Italy." It's a modern bustling city (albeit with some very old things, like paleo-Christian churches), and feels more "European" than "Italian." However, one plus is that, unlike Venice and central Florence, the native to tourist ratio is heavily skewed in favor of natives (although like lots of big cities, Milan residents are often from somewhere else). Another plus is that, even though it's large, it's much easier to get around than many smaller Italian cities, due to the very efficient and extensive Metro.

Just to note that your visit to the Last Supper is limited to 15 minutes, so yes, some people would have a hard time justifying a trip for mostly that one thing. Milan is a fine city, it's just less attractive than Venice, Florence or Rome. If it were me, I'd cut a day off Lake Como, and stay one night in Milan, to see those things, since you have to pass through there anyway.

3 nights in Menaggio would be good and probably more than adequate. Coming from Lugano it’s a straight shot to Menaggio, about 45 min by car (along lake Lugano) or maybe one hour by bus. Two nights might be sufficient, but it wouldn’t be as relaxing. Two nights is only one day, and I’m sure you want to visit all 3 towns (Menaggio, Bellagio, Varenna) and maybe take some relaxing ferry rides.
Lugano also probably needs no more than 3 nights.
I freed up two nights in your schedule already. Spend them in Milan, if you wish.

From Lugano (worth one day at most), the small, pretty village of Morcote is close by. Lovely gravesite and hikes up from the main road. Bellinzona is well worth a day trip, three castles (Unesco) and a nice central area. Seeing as you have a car, Avegno, Brisaggo and Locarno to the west, the Chestnut Trail through three or four villages (we got lost).

Varenna and Bellagio are both more attractive villages than Menaggio. I would skip Milan ( we missed our Last Supper), but that's only my way, you have to decide for yourself. Enough to do around the lake if you like to take it easy. Greenway Walk on the west side, Lenno has a very nice market day (Thursday?), many villas and gardens. I was not fussed about the town of Como itself but nice enough for the ferry ride.

I spent a day in Milan and was glad I did. The two highlights for me were The Last Supper and walking on the roof of the Duomo.

I have stayed in Varenna twice arriving by car once and then by train from Milan.I went to Il Caminetto for dinner twice and just loved the food and service and they pick you up at the ferry stop in Varenna,beautiful views of the lake on the way up to the Restaurant.I personally enjoyed Milan with its Cathedral,Last Supper,La Scala,Sforza Castle and the Mall which can be done in a day.
Mike