Venice 3 nights, Florence 3 nights, Amalfi coast 4 nights, Rome 4 nights.
That's the order and amount of days I'd give each location if committed to doing all 4, and having 14 nights on the ground to work with. If you only have 12 nights on the ground in Italy, I'd cut the Amalfi Coast and save it for a future trip. That trip would look like:
Venice 3 nights, Florence 5 nights (including 1 or 2 day trips), Rome 4 nights
Most of us put Venice on the front end as flights out often leave very early in the AM; can cause some issues getting to the airport, and one has to get up at a lousy hour. Rome goes on the back end; should you hit a transport snag, the coast is not where you want to be the day you're flying home.
I'm a big fan of Florence and too often see new visitors cut it short. Then again, how much someone else will enjoy all that she has to offer depends on their interest in Renaissance art and architecture. We had no issue filling the better part of a week there with no day trips at all. Still, it is a great base for jaunts to Siena, Lucca, Pisa and some other locations so one doesn't HAVE to be the art dork that I am to settle nicely in for longer versus a shorter time. :O)
How to get around? Train, no question. Italian trains are comfortable and very efficient; you can generally get on one in the middle of one city and get off smack in the middle of another. If you can commit to a specific day and time to travel, you can also get some nice price breaks on tickets for the fastest of them if purchasing in advance. The "fast" trains are definitely how you'll want to get from Venice to Florence, and Florence to Rome. If taking day trips, there are a few exceptions, such as Florence> Siena, where bus is the recommended method.
If you keep the Amalfi, the better transport options depend on WHERE you are staying on the Amalfi or Sorrentine coasts; could be a combination of fast trains, slow commuter train, ferries, private drivers and/or buses. Whatever you choose, unless you're staying in Salerno, there is no DIRECT transport (meaning no change from one type of conveyance to another) to the Amalfi Coast via public transport.
This is a great introduction to train travel in Italy; highly recommend spending some time with it although I don't recommend using it to order tickets; we'll steer you to the best sources for that once you have your itinerary down.