With only a very short 10 day time frame, I would also suggest leaving the Amalfi Coast for another visit. While it might be feasible, it will require more travel time as transit isn't as easy in that area. The Cinque Terre is not entirely the same but it fits better with the circumstances in this case.
What time does your flight arrive on 19 April? There are lots of ways you could arrange this trip, and this is one suggestion....
- Apr. 18 - D1 - Flight to MXP
- Apr. 19 - D2 / N1 - Arrive MXP, train to Cinque Terre (it's a good place to recover from jet lag for a few days) - I'd suggest staying in Monterosso, as it has the best choice of accommodations).
- Apr. 20 - D3 / N2 - C.T.
- Apr. 21 - D4 / N3 - C.T.
- Apr. 22 - D5 / N1 - Train to Roma Termini (about 4 hours or so)
- Apr. 23 - D6 / N2 - Rome
- Apr. 24 - D7 / N3 - Rome
- Apr. 25 - D8 / N1 - Train to either Venice or Florence (Florence offers some good day trip possibilities such as Lucca or Siena).
- Apr. 26 - D9 / N2 - V or F
- Apr. 27 - D10 / N3 - V or F
- Apr. 28 - D11 / N1 - Train back to Milan
- Apr. 29 - D12 - Flight home
I would highly recommend packing along a copy of the Rick Steves Italy 2018 guidebook, as there's an enormous amount of information there that will help your trip go smoothly. One of the things you'll have to get up to speed on is the format used for meals, with Primi, Secondi, Contorni, Dolci, etc.
As this is your first trip to Italy, there are a few potentially expensive caveats to be aware of when using trains and other public transit. If you need more information on that, post another note. You'll also need to know the specific stations you'll be using in Italian when buying tickets.
Have you researched things like how to travel from MXP to Milano Centrale? Have you been to Europe before? If not, you'll need to consider things like Plug Adaptors, travel with smartphones, obtaining cash, etc. Are you planning to wear Money Belts? Have you checked the expiry date on your Passports?
Have you got any hotels booked? I'd suggest doing that as soon as you finalize your Itinerary. The group here can offer suggestions on each of the locations you plan on staying.