Como, Lucerne and Amsterdam

Get the 3 day Swiss Travel Pass and catch the Bernina Express scenic train in Tirano on the way to Luzern go up Mt Pilatus early only and then Mt Titlis on a day trip from Luzern; and then fly out of Zurich airport to Amsterdam; it is about an hour on the train direct from Luzern.

Forget the car and leave the sightseeing to the train. Especially in Switzerland.

From the Milan Airport get to Milan Central rail station. Switch to a regional train to Varenna. (Varenna-Esino) Stay in Varenna. Varenna is a town on Lake Como and the ferries will allow you to explore numerous parts of the lake. From there, take the train north to Tirano. Cross the street from the Italian train station to the Swiss train station. (Both stations are very small.) Take the Bernina Express to Chur. From there, transfer to another train to Lucerne. From Lucerne you can tak a train direct to the Zurich Airport without having to change.

Train schedules for September are not out yet but should be about three months ahead of time. For the Bernina Express you will need to book months in advance--if it's running.

Before buying a swiss rail pass compare ticket prices using the Swiss Half Fare Card. I saved money doing that.

Flying into Milan from the US, you will arrive in the morning, so two nights will probably be enough, giving you half a day to settle and a full day to visit and enjoy the city (which I find to be enough, the center is very compact).
That extra night could be added in Amsterdam, allowing for a day trip somewhere (tons of options: Haarlem, Delft...).

Regarding the Swiss half fare card: it seems unlikely to bring significant savings, maybe $10-15 over the full fares. There are "super saver" advance fares that can bring better savings; they are available up to 60 days (or maybe 90?) before departure.

If you do arrive in Milan in the morning, you likely wont be able to check into a hotel yet, so do consider going onward to Lake Como. Where do you plan to stay there?

Definitely would not miss The Last Supper at Santa Maria delle Grazie in Milan. You may have to book a walking tour to get a ticket but it's well worth it! Here's an idea that may be worth exploring. Fly into Milan and rent a car to visit the city and the Lake Como area. Then fly from Milan to Zurich and rent a car to visit the Lucerne area. Then fly from Zurich to Amsterdam. Renting a car in Milan and dropping it in Zurich will likely be expensive.

You don't need three nights for Milan, but it is worth two nights. From there you can take a one-hour non-stop train to Varenna which is where I would sleep to explore Lake Como. I would not rent a car in Italy and drop it off in Switzerland because it is very expensive to do so. The cost of renting a car in one country and returning it in another is similar to renting a vehicle in one state and dropping it off in another.
To get to Lucerne you can take a train to Lugano that requires one connection. Total time takes 2 1/2h. From Lugano take the scenic Gotthard Panorama Express to Fluelen and a boat to Lucerne. I've spent four nights in Lugano back in '81 and liked it so if you need to sleep there there are much worse places.
To get to Amsterdam you'll need to take a non-stop train to Zurich's airport (1h + 15m) and Swiss Air offers flights as low as $110 that will take about 2h to get there.
Sounds like a nice trip.

I don't know why somebody would fly from Milan to Zurich and then backtrack to Luzern when it is faster, much more scenic, and so much easier to take a train.

I don’t know what your interests are, but personally I wouldn’t spend any time in Milan. Upon arrival I would take the train to Varenna on Lake Como and spend 3 nights there. You can explore the lake by ferry. A car would be a hindrance and as others have said, it is very expensive to rent a car in one country and drop it in another. Count how much time you have by how many nights you will be there. For example 3 nights is 2 full days, etc. It will take more time than you think to change locations. I’ve been to Lake Como twice, each time spending 3 nights in Varenna. In my opinion that was sufficient time. I can’t advise you about Switzerland, but you’ve heard from others. Flying from there to Amsterdam is a good plan. I would take the 2-3 nights you save by not staying in Milan and add them to Amsterdam. Enjoy your planning and your trip!

Frank II and Nigel are giving the good advice — the train is what you want, not all this driving and/or flying.

Fly into Milan and rent a car to visit the city and the Lake Como
area. Then fly from Milan to Zurich and rent a car to visit the
Lucerne area.

No offence, but this is really bad advice. Milan is, like all European cities, not a city that is best visited by car. Also, flying from Milan to Zürich to get to Luzern is just crazy. There are direct trains from Milan to Luzern in 3:30, trains that also stop in Como (the town). Flying takes longer, cost more and is less comfortable.

Their post said "car and driver", so they aren't thinking of a rental car to drive themselves.

They are much better off to take the train to Varenna on Lake Como, as advised by many above. The trip by direct train takes just over an hour and used to cost €6. I did not check the current price.

Varenna is a great place to stay on Lake Como; a charming little town with several hotels right on the lake. We like Alberto Milano, staying in the top floor room in the annex, with a large balcony overlooking the lake. Lots of stairs so this room is not for everyone.

After 3 nights at Varenna they can board the northbound train to Tirano and change there tona Swiss train going over the Bernina Pass. This is the most scenic of the train routes between Italy and Switzerland by far---all the others go under the Alps in tunnels at least part of the way, and some for much of the way.

Regular regionalmtrains ply this route every hour, or they can ride the special Bernina Express train, with its panoramic windows. See photos and get information on tickets here:

https://www.rhb.ch/en/panoramic-trains/bernina-express

We have only ridden the regional train and it was fine, but I cannot compare it with the named Bernina Express.

If they drop a night in MIlan (which they should) they would have time for an overnight stop in the Alps on the way to Lucerne. Maybe Pontresina? St. Moritz? A mountain inn with its own stop on the train line? It's all good.

Then continue to Luzern for the 3 nights there. From Luzern to Amsterdam they could tKe the train, it I would suggest they fly from Zurich ( one-plus hours by train from Luzern, with trains going right into the airport). We have taken the train from Amsterdam to Switzerland ( Basel) and it was a long ride; comfortable enough but not all that scenic ( compared to Switzerland).

Beware of those who give advice and are critical of the advice from others - especially if they haven't made the trip. I've driven from Milan to Lucerne to Zurich. You can drive from Malpensa to Como in 45 minutes - it'll take 3 hours to take the train/bus/taxi from the airport to the Milan train station to Varenna (yes I've taken the train between Malpensa and Milan and Milan and Varenna) unless you have perfect timing. Varenna's great but what if you prefer another base on the lake? When I travel I research all my transportation options to hopefully achieve the ideal balance between time, flexibility, cost, fatigue and reliability. If I was planning your trip I would look at all your driving and flying options as well as the train. BTW - a year ago we flew to Paris and were going to take the train to Colmar but a transportation strike resulted in our driving instead, and a few years before that we got stuck in Greece because of an air traffic controllers strike. Researching all your transportation options in advance also allows you to have a backup plan in case of surprises.

@KBK: The OP wants to spend some time in Milan before going to the lake. So the question is how to get from Malpensa to Milan, and then a few days later from Milan to the lake. Also, comparing driving time to Como to train travel time to Varenna is not really useful.

@Badger: I'm simply trying to give Easy Traveler some ideas to consider. Here's another one - fly into Malpensa and immediately drive to Lucerne (less than 3 hours) and them loop back to Italy before flying from Milan to Amsterdam. If it was my trip I would try to find the most efficient, cost effective way to visit the four places and be flexible on the order of those places. I might even change the places I want to visit based on my research. What you see from a train you can sometimes touch by driving.

so KBK, two trips through the Gotthard Tunnel?

Why backtrack?

I want to thank everyone for their varied suggestions. I appreciate ALL of them very much. It has given me many options I would not have thought of. I will definitely research them all and choose what is best for me. Once again, thank you.

I agree with Nigel, why add a lot of backtracking to a trip?

Looping back say via Chur and Varenna would be much more fun and scenic IMO than simply backtracking. I always try to avoid backtracking but sometimes a little partial backtracking makes sense if it saves a lot of time.

I would never advise driving for 3 hours after an international flight. That sounds very unsafe to me.