Switzerland and Italy

Go in May because mountains make their own weather and visibility may be best in May. It’s best to allow five nights for the Lauterbrunnen Valley for this reason. I wouldn’t spend time in Zurich other than using it as a transportation hub. The old town is nearly depleted and there’s not much to see or do in Zurich. The Berner Oberland is Switzerland’s true highlight.
I’m not sure when ferries will be running on the Amalfi Coast but I know all modes of transportation will be operating in the month of May. Hotels booked through a third party i.e., Booking.com, etc. are cheaper the further out booked. What I like about Airbnb’s is you can get one with a washing machine making it convenient to do laundry.

If you go in March, plan for snow in the Alps, so I would pick May. Here’s a trip report that covers some of the places you’re going, plus a few extras. It’s almost ten years old, but most of it still is relevant. (And I’ve stayed in Stresa several more times since that trip - love it!). Now I stay on Isola Bella the last evening I’m in Stresa at the small Boutique Hotel Elvezia.

https://community.ricksteves.com/travel-forum/trip-reports/trip-report-switzerland-italy

Depending on what you want to do in the Alps and AC, March may be too early. Hiking trails in the alps will likely still get closed, and I don't think the ferries are running on the AC. May will be better in that regard. It will be busier then, obviously, but not as bad as the summer months.
Types of accommodations are a personal choice. We prefer hotels with a 24 hr front desk in cities, but often opt for a b&b in small towns and villages. I have a problem with Airbnb, so dont use them, but have occasionally done short term apartment rentals when staying in one place for more than a week.

A lot of Swiss lodging may be shutting down the end of March, depending on when ski season is closing.
Some Alpine lifts don't open until late May or early June.
Avoid Easter and the surrounding weeks in Italy. Easter is late March this year. Not sure they are any "deals" left for May in Amalfi- most folks book 8-9 months out. There really isn't a shoulder season in the Amalfi area anymore- spring and fall are packed.
What are your priorities?
Safe travels!

Thank you all for your responses. Thank you, Jean, for the itinerary, this will be a great help. It looks like everyone agrees on May being a better bet to go to both Switzerland and AC. I really appreciate all the advice!!!

Setting aside the ethics of Airbnb (I'll let you decide that for yourself), I will note that Switzerland can be very expensive. And if you are American, the dollar is taking a shellacking right now against the Swiss Franc. One cost saving measure is to have your own apartment/kitchen space and cook meals in. Raw ingredients from the grocery store, or the local markets, will be much cheaper than restaurants and bars. Of course, you shouldn't skip dining out entirely -- that's part of the experience! But with hotels you don't have much choice except to go out, while with apartment rentals you can dictate the options.

If you go in May, I would start in Italy and then go on to Switzerland for the weather. As far as better deals, summer is always going to be more expensive and more crowded than spring, at least it used to be. Lots of things to consider.

How fun! We are also planning on going to Switzerland and AC in late April, early May. I have a post in the forum titled "best home bases for Switzerland in April" with so much great feedback. Sadly, many people are recommending I do Lake Como or elsewhere instead of AC because it is in a whole different country and will take some time to get to. I am just curious, how many days total do you have for your vacation and how long are you planning on spending in the Amalfi coast? I've read that 2-4 days is enough. Just curious what your thoughts are.

CamNM we plan on 14 to 16 days for the whole trip. Hoping for 3 to 4 days in AC. We have been told the same thing regarding the AC and that we should stay up in Lake Como and the furthest Rome. I’m still trying to figure things out. Our thoughts were to fly into Zurich and leave out of Naples.

I'll comment here on my own itinerary in late May that tackled these locations (several years ago). We flew from Midwest US to ZRH. Stayed in Zurich 2 nights, stayed in Bern 4 nights. Flew ZRH to FCO for 4 nights, then trains to Naples-Sorrento and bus to Positano where we stayed 3 nights. We then flew home from Naples. We used hotel points for all nights in Zurich, Bern and Rome. We had a Swiss Pass, and spent time in Bern, Thun, and some very small towns that are ancestorally important to my husband's family. I felt like we did a lot, I liked our trip and felt good about the pace.

I'd lean more towards May, OP.