Lucerne to Milan and Bellagio (Road trip/Day trip)

The train system is very efficient, and I would recommend not using a car. This would be too much for me as a day trip.
If you're going to Lake Como, slow down and relax and enjoy the beauty. Why are you going to Milan? Also, Milan has traffic.
Safe travels.

ViaMichelin.com estimates the driving time (no usage of lake boats) at 7 hr. 15 min. That is obviously far too much driving when you intend to see two places along the way, Bellagio and Milan. I am doubtful that tossing a boat into the mix would save time, if only because the scheduled boats don't run all that frequently and you'd have to pad your schedule to allow for that, but cannot say for sure. If by "water taxi" between Como and Bellagio you mean a private transfer, I'd expect that to be very costly.

Viamichelin say 2 1/2 hours to Menaggio. Park the car there and take the shuttle ferry to Bellagio. Along with other spots on the lake accessible by ferry, i think you'll have a full day skipping Milan.

Thanks, everyone!

I'll likely drive to Menaggio and then ferry to Bellagio and maybe add another town. Milan seems way out of the way and not possible to include in the day trip.

check to make sure that the Gotthard tunnel is clear - all 9 and a half miles of it before you set off in either direction. There are other, but very very slow alternatives if it is bottled up.

remember that the mid lake ferries go in a circle. You may find that your Menaggio to Bellagio ferry goes to Varenna first.

If for any reason you decide to use the Gotthard pass instead of the tunnel, check first if it is open. It usually opens mid-May, depending on snow conditions. I remember around May 20 one year, a few days after its opening, we were driving between two walls of snow on top, but the road itself had been cleared. I found it quite spectacular.