Up to 15 days in Italy -April 2016

We could give better advice, if you give some indication of your interests. Also, do the 15 days include your flight days? In general, I would say you have too many destinations and are asking for more. E.g., 3 - 4 nights in Venice with 2 possible day trips.

The drawback of not having a car is that you are tied to train schedules, and you have to consider the baggage storage problem. It makes stopping off on the way to somewhere a lot more complicated.

A few years ago, we drove from near Florence to Sorrento. As I recall, we looked at the train option and decided against it. But then, we were needing to get to Florence first. I don't think you can do it without changing trains, but that's an easy one to check. At any rate, it's not a short hop. I wouldn't think of wondering about what I could see on the way.

You've already got 4 destinations listed for Sorrento; the town itself, Amalfi Coast, Capri, and Pompeii. Each is a full day in itself. If you're going to do that, you need 5 nights there.

In my opinion, your next step should be to research how long it will take you to get from place to place, including your proposed day trips, e.g., Venice - Dolomites. You'll get a better sense of what is feasible. You probably can't get schedules for next April; but if you put in dummy dates, you can at least get a some idea of timing.
Good thing you are getting an early start on planning.

Several years ago I took the train to Padua from Venice enroute to Bologna. There was a baggage deposit area at the Padua train station. I left my luggage there while visiting the Scrovegni Chapel and St. Anthony's Basilica. It worked out perfectly. Sorry, but I didn't make it to Verona. Hope you have a great trip.

Padova and Verona train stations both have bag storage. If visiting Padova, it's recommended to reserve ahead for the Scrovegni Chapel; last September, I did it the day before. When you're done there, you can simply buy your onward train ticket at the station for the next available departure (full fare of €40 per person in 2nd class to Florence). Or, if you want an advance-discount for that train ride, then the ticket will be booked for a specific (presumably afternoon) departure time.

Padua is a good place to spend a day en route from Venice to Florence.

Verona is one of my favorite towns in Italy, lots of sights and good ambience in the evenings (which Padua lacked for me). I'm not familiar enough with transportation, but perhaps Verona would make a better place for a 2-night stay and a day trip to the Dolomites. If not, you could see a lot of the Veronese sights on a day trip from Venice.

It's a long trip from Florence to Sorrento and you'll have luggage with you. Just go straight from hotel to hotel, if possible take a train that goes to Naples without a change in Rome.

If your choice is 3N Venice and 5N Sorrento, or 4N Venice and 4N Sorrento, I guess it depends on whether you want to see Verona. If you can add a night to Sorrento without subtracting from Venice, do so. Sorrento is lovely and there are lots of day trips.

Don't pay Venice prices for time when you plan to travel outside Venice - move to Padua.

Venice (3 nts)
Padua (3 nts) - day trips to Dolomites and Verona
Florence (2 nts)
Sorrento (4 nts) - day trips to Pompeii+Naples / Capri / Positano+Amalfi
Rome (3 nts)

Since your grandmother came from the Dolomites I would strongly suggest you stay there morethan a short day. It is amazing how welcoming these people, especially from the smaller towns, and are so glad to see someone come back to see their roots.

I had that experience about 8 years ago. I have been back once and am planning to go this Sept. And I am planning to spend two full days there because I have kepet in contact with several people and I want to see them again.

I would really think over your stay in the Dolomites.....

Thank you for your great suggestions! We're going to rely on the train rather than renting a car and may have a driver for some of the time in Sorrento. In 2011 my daughter backpacked throughout Europe and I joined her for 18 days. Other than flying from Prague to London we took trains everywhere. We will think over our time in the Dolomites and possibly staying in Verona. Thanks again!