For 10 days I'm not sure I'd try for more than 2 stops--every time we've tried to go for 3 some kind of travel hassle has gotten in the way. When we went to Lake Como a few years ago we were trying for 2 nights outside Turin, 3 in Varenna (mid-Lake Como), and 5 in Venice. The Turin part was for a "finding your roots" visit and we never made it thanks to flight cancellations--and this is before COVID!--so we had 4 on Lake Como and were glad to have the extra day.

Lake Como is absolutely worth your time and we loved Varenna. It's the best connected village via rail. A car is pretty worse than useless for traveling around the lake. It's got a wonderful ferry system (schedules here: https://www.navigazionelaghi.it/risultati?lake=Como). With a car it seems as if Menaggio would be the best town for your base, just based on the way it's laid out, and it's got great ferry connections to Bellagio, Varenna, etc. Bellagio would be pretty awful for dealing with a car and inconvenient for going to Lugano--and we didn't love it in general. You hate to say that something is too touristy when you're a tourist, but it is REALLY touristy, and not any more beautiful than any of the other beautiful villages.

Spending just two nights around Lake Como kind of means you're not getting the best of that particular area--which is to wake up, look at the lake and the mountains while having a cappuccino and cornetto, take a ferry, maybe visit a garden or villa, have a long lunch while looking at the lake and the mountains, ferry back for aperitivo while looking at the lake and the mountains, and then figure out your dinner plans--which should include views of the lake and the mountains. It's perhaps the most relaxing place on earth, and is perfect for just unwinding and not having a list of things to do.

In terms of other stops, the Freccia and Italo trains (or a car, I love driving in Italy) make anything down to around central Tuscany feasible. Places I'd like to go sometime soon that we haven't visited before include Turin, the surrounding areas, and the Val D'Aosta; the other lakes (especially Lago di Maggiore and Lago di Orta); Bologna, Ravenna, Parma...just make sure you stay your last night in Milan or near Malpensa because that airport, and getting to it, can be a real beast.