How to spend 15 days in Italy + March/April weather concerns?

Welcome to the forum!

It’s a decent plan but I would add a night to Venice and a night to Rome
Spend the rest in Sorrento or Salerno

You’ll probably need to spend night before departure in Naples

Note that Easter is March 31

That is the time to travel. Less people.
Unless you are a beachgoer there is nothing in your way. Just have a jacket.
That is our preferred time of year.

Didn't take Easter into consideration! Will businesses be closed in Rome/other cities, does it get chaotic?

The one change I would make is to fly into Venice and home from Rome. Rome would give you many better options for returning than Naples. Second, it is easy to get to the Rome airport if in Rome. Naples can be a super pain especially if you were staying in the Sorrento area. (Positano is out of the question.)

I would not worry too much about the weather but plan for some rain. Our experience from time period is light showers maybe for 20 minutes, sometimes off and on during the day, but no days where it rained all day. It will be cooler but it is still a good time to go.

I haven't been to Italy over the Easter period, but it's a popular time for folks to travel when they want to avoid high heat and peak-season crowds. Therefore, the most popular cities will be heavily trafficked (affecting hotel rates), and you must be aware of the need to grab your key sightseeing tickets early. The two that immediately come to mind are the Colosseum and Vatican Museums in Rome, but there will be an issue or two in Florence as well, I assume.

I consider 2 nights in Venice inadequate, especially when it's your landing point in Europe. You may be sleep-walking zombies on your arrival day, leaving just one real day to explore Venice. That's a problem, because it most likely means you'll spend a great percentage of your time around the San Marco and Rialto areas along with all the day-trippers who what to see those same sights. (Have you seen photos of what small port towns look like whan a large cruise dip arrives?) You need time to wander around the back streets in Venice to appreciate its charm--maybe hop on a vaporetto over to Burano, just as one example.

Three nights in Rome is also a very short visit.

I think with 13 nights (12 days) you're spreading yourselves awfully thin by trying to travel from Venice all the way to the Amalfi Coast. I love Venice, but I'd tend to drop it this time so you have more time for your other stops.

I agree that your outbound transportation options will be better if you travel from Florence to Naples/Amalfi Coast and then to Rome. Otherwise, your last full day in Italy is likely to require relocation back to Rome for your flight departure.

I would not be concerned about the weather. We spent the first 2 weeks of February in Florence, Rome and Naples and the weather was nice for sightseeing. We dressed in layers. We only had one rainy day. You’ll have warmer weather than we did — so fewer layers.

My original plan was to go to Naples and Rome, but it was cheaper to fly to Florence and Naples, than to Rome. Weird, but that’s what it was. So I added Florence. We had 3 nights in Florence (not enough time), 7 nights in Rome (about right) and 3 nights in Naples (not enough time). Of course, not having enough time is a common problem. We were happy with our itinerary. Florence was a bonus for us, so even too short of time, was more than we had originally thought we would have.

I think you might enjoy your itinerary more if you cut something. It sounds like you have 4 destinations: Venice, Florence, Rome, Amalfi. If this is your first visit to Rome, you need more than 3 nights. I think if you dropped Venice or Amalfi, you would have a nice itinerary that is not too rushed.