Northern Italy

Assuming you have 12 nights:

Varenna and Bellagio are both in Lake Como. I would stay in Varenna for 3-4 nights and visit other points on the lake like Bellagio.

Lake Garda has several spots folks here on the Forum like. Malcesine, Limone, Sirmione. We found Lake Garda a bit boring after two nights there. May will not be "high season."

If you spend 4 nights in Varenna and 2 at Lake Garda, you have time for a nice stay in Tuscany or the Cinque Terre. Perhaps 3 nights in a Tuscan town and 3 in one of the 5 villages?

Use www.Rome2Rio.com to estimate transfer time between locations whether by train, bus, or driving.

I took the train into Varenna and ferried across Lake Como and spent 3 nights in Cadenabbia. I found an AirBnB right on the lake by the ferry dock. The sunrise is beautiful from that side and you have great views of Bellagio and Varenna. In my opinion, I would stay in either Cadenabbia or Menaggio. There is not much to do in Cadenabbia except for walk around their gardens and enjoy the views but it's alot cheaper and easy to ferry from. Menaggio is larger and you have access to the bus line to Lugano, Switzerland (I think it's C12?) if you want to head there for a day trip. I went to Bellagio and ended up only spending a few hours there since it's mainly shopping and food. You can buy ferry passes that will let you ferry town to town for the day an unlimited amount of times. I used that one of the days because I just loved being on the lake. I didn't do much walking around in Varenna...just from the train station to the ferry, but there is a tourist office there that can help you with planning day trips and get train tickets. From Varenna, you could do day trips to to cities like Verona, Venice or Milan if you want to venture off the lake. If you have 12-13 days, I think you plenty of time to visit the Cinque Terra or Tuscany. It just depends on if you want to do some day trips from the lake and how many.

3 nights each lake is adequate. Choose one place per lake.
Lake Como is best at midlake (Varenna, Bellagio, Menaggio).
Lake Garda is best at North Lake. Malcesine is a good base. From there you can visit Limone, Riva, etc. via boat. And you can go up to Monte Baldo with the cableway for a great view. You can also visit Sirmione via boat or bus.
Tuscany (Florence) is about 3-4 hours from either lake. Cinque Terre requires longer times.
From lake Como you catch the train in Varenna to Milan, then proceed wherever.
From Malcesine you need a bus to Verona, then the train.

Are you flying in and out of Milan, or where? It basically sounds like your plan is realistic, but I'm not sure how much time you need to allow to get to and from your airport(s) of arrival and departure.

To the last point raised, if you land in Milan in the morning or early afternoon, you can take the train to your first lake destination. From Milan (not the airport) to Varenna is an hour's ride. From the CT, take and late afternoon train to Milan for your last night so you have easy access to the airport. It's 3-4 hours from Monterosso to Milan. Or skip the CT and spend your time in Tuscany - Florence or Siena, depending on what you want to do. There are frequent fast trains between Florence and Milan. I keep saying Milan because you will probably get the most flight options and best deals. Florence can be pretty crowded and Siena takes longer to get to/from (not on a major train line). Instead of Tuscany, consider staying in Bologna, which is a rail hub, and not a big tourist destination. It's a very pleasant town with some "second-tier" sights, but the big draw is the ease of taking day trips to other great places - Modena, Ferrara, Padua, Ravenna - and of course, from the lakes and to Milan.

We went to Lake Como and Lake Garda this past year. I much preferred Lake Como and not sure if you really need to go to Garda unless there is something specific you want to see there. May is supposed to be the height of beautiful flowers in the Lake Como area. If this interests you, there are several homes/gardens that could fill up a couple of days there.