Thoughts on filling a week in Italy...

I'd just do Milan as a long day trip from Varenna (Lake Como area). There is well manned left luggage service at the Milan Central train station. From Varenna you might head east to Verona or Lake Garda area. This would you give you a lot of options like a trip to see the Dolomites. Then go to Florence.

If Amalfi is what appeals most to you then go there.
I would take a night or 2 at least from Milan/Como and add to your 4 nights so you have 6 in Amalfi/Sorrento region.

You can take a direct train from Florence to Salerno (3.5-4 hours)- then ferry to your Amalfi coast town choice- we preferred Amalfi for it's less expensive lodgings, more "real" feel and good transportation.
Spend 3 nights in Amalfi- visit Ravello, Positano from here
Then move to Sorrento for 3 nights- visit Pompeii, Capri and /or Naples from here
Return to Rome for your final 4 nights.

You definitely don’t need three nights in Milan which is a very doable day trip from Lake Como by direct train (>2h).
I agree with ChristineH about taking a direct train from Florence to Salerno and then a ferry to either Amalfi or Positano and sleep there. Positano will put you in the middle of the coast making Pompeii easier to get to.
I stayed in Sorrento when I visited the coastline and didn’t like it, however, Sorrento is situated well on the Amalfi coast making it a convenient transportation hub.
If you stay in Sorrento and go back to Rome via Naples then you have the option of taking the Circumvesuviana commuter train that runs between Sorrento and Naples. Pompeii is a stop on this route. This train is fine if not transporting luggage.

One of the problems with commuter trains is you get a lot of pick pockets riding with you. When you are hauling valuables take the ferry from Sorrento to Naples and a taxi to the train station. If you stay in Positano or Amalfi take a ferry to Salerno and a direct train to Rome (2h 15m).

I would agree with all of the above recommendations. 2 nights in Milan is sufficient. While in Lake Como, since you are there so long, consider a day trip to Lugano, Switzerland. You can either get the bus out of Menagio or ferry all the way, or do one way bus and the other by ferry. For a visit to the coast I would suggest staying in Sorrento. From there you can venture south to Amalfi, Positano, Ravello or north to Pompeii and using that as your home base you don’t need to worry about where to store your luggage when making visits along the way. Take the hair-raising, white-knuckle bus ride to Amalfi, sit in the front seat on the right side of the bus to fully appreciate the skill, and nerves of steel, of the bus driver. Be particularly alert to pickpockets on the Circumvesuviana train. Have fun! We are taking our 12 year-old grandson to Europe in June and Italy will be on our itinerary as well. So looking forward to being able to travel again.

Thank you all very much! This is just the kind of information I was looking for. If anybody thinks of anything else, chime in.

We wished we hadn’t taken time from Lake Como to go to Lake Lugano on Switzerland. We preferred Lake Como and regretted taking time away from it.
From Milan it is an easy drive to Santa Margherita Ligure and/or Camogli. You can drive, take a boat or walk out to Portofino if interested. We have gone there several times and always enjoy this area. Very different from the CT. We liked this area much more than the Amalfi Coast.

I second the suggestions for both Santa Margherita Ligure/Positano and Amalfi Coast/Sorrento.

Pompeii is an Italy must-see for me. Consider to visit Capri from Sorrento if the time and budget allow.

consider a day trip to Lugano

Have to agree with Suki after spending four nights in Lugano. It has a very small old town and the lake is pretty but Como is better.