I think Rick rates the mid-lake ferry trip (Bellagio, Varenna, Menaggio) as the #1 sight/activity on the lake, and I tend to agree with that. The views from the lake are great, and the ferry trip isn't terribly expensive or time-consuming. Villa Balbianello and its gardens are a good option. I'm not big on stately homes unless they were decorated in the style of Art Nouveau or Art Deco, but I can understand why others really enjoy them.
As it happens, my favorite activity on the lake was walking part of the Greenway. I was there as August turned into September, so it was still a busy period, and I don't think I saw more than 2 other people (and they were not tourists) during the 2 or 3 hours I spent following the Greenway. I felt sort of like an explorer, walking through a few of the little settlements above the lake. I have to warn you, though, that it may be a bit tricky to find a place where you can join the Greenway. Signage is a bit spotty. I first tried to pick it up in the area around Tremezzo, but I wasn't sure I was in the right place, despite having a paper map of the Greenway and an electronic map on my phone. I ended up hopping on a bus down to Colonno, the southern end of the Greenway, the next afternoon. That worked well, because you can see from the Google Map exactly how you need to walk (quite uphill) to get from the bus stop on the road along the lake up to the Greenway. I had a few moments of uncertainty during my walk, but there's nothing to be concerned about, because you always know where the lake is (even if it's screened by vegetation)--it's downhill! Buses run along the lake road hourly, as I recall, so you can always follow a path heading downhill and get to a bus stop (marked on Google Maps) without difficulty. The key think will be to have the ferry schedule at hand and not plan to take the last ferry back to Bellagio.
Try to find the summer ferry schedule online and print it out before you head over to Italy, because you may have trouble sourcing a schedule leaflet once on the ground; I think the schedule changes on June 1.
Here's where to look: https://www.navigazionelaghi.it/en/tickets-and-timetables-lake-como/
I'd recommend checking with hotel staff about the late-afternoon options back to Bellagio. It can sometimes be a bit difficult to interpret Italian public-transportation schedules (though this one is clearer than most).
