Hi and welcome to the forums!
The not-so-good news first? You have too many places on your wish list for 13 days of sightseeing so you're going to have to pare down. Every time you make a move you're going to kill about a 1/2 a day in the re-location process, and you're looking at 5 moves: that's 2.5 days you'd lose to trains/stations, packing/unpacking, checking in and out... all that eats TIME.
OK, now that we've gotten THAT out of the way, it's great that you're flying in and out of two different cities! If you are flying into Venice, the two locations I'd scratch are the outliers: Milan and Amalfi Coast. The Amalfi needs more time that you can give it and would involve backtracking to Rome. Milan? I'm OK with it but think your time is better off in some of the other cities. I'd consider:
Sat 25: late arrival in Venice - this does not count as a sightseeing day
Sun 26 - Venice
Monday 27 - Venice
Tuesday 28 - Venice
Wed 29 - Florence (partial Day)
Thursday 30 - Florence
Friday 31 - Florence - day trip to Siena or Lucca
Sat 1 - Florence
Sun 2 - Cinque Terre (partial day)
Monday 3 - Cinque Terre
Tuesday 4 - Rome (partial day)
Friday 5 - Rome
Sat 6 - Rome
Sunday 7 - Rome
Monday 8 - fly home from Rome
This could be shuffled a bit to take a day from one city, like Florence, to add to another you might prefer to spend more time in. I consider 4 nights the minimum for Rome, and art lovers can find a LOT to see in Florence. That one also makes a good base for a day trip or two, time allowing. Some of our posters are passionate about Venice and you could still be a little foggy on your first full day so I gave you 3 of those to get on your feet and explore maybe one of the smaller islands as well as Venice itself.
Just a suggestion?
Editing to add: TRAINS are the way to go, no question. They'll get you from one place to another the fastest, and you do not want or need a car in Italian cities; you will be exploring them on your feet (lots of pedestrianized areas) and sometimes with public transit.