Brent- As your subject line says- this is your FIRST trip to Italy. Tell yourself you will return and don't try to cram in so much in this trip or you probably won't even WANT to return ;)
10 nights is just barely enough for 3 major locations and since you are flying in/out of Rome I'd stick with the southern locations. Venice needs minimum 3 nights to really experience all it has to offer. Most folks that don't like Venice only give it 1 or 2 nights.
Milan is an easy place to fly in/out of on a future trip- that's when you can see Milan, Lake Como and Venice.
Rome needs at least 3 nights- that’s the bare minimum. You could easily spend 10 nights in a Rome!
Florence should get 3 - add a night for every day trip. If you are not big into art you can get away with 2 but keep in mind if you go there on arrival that first day is jet lag and late arrival
A 2 night stay is really only 1 day for being/seeing where you are. Each time you change locations you lose a half a day or more- packing up/checking out/getting to train/train ride/getting to new lodgings/checking in/getting oriented.
I'll second Kathy's suggestions- but go to Florence on arrival for the reasons SJS has stated. It's closer/easier.
Fly to Rome-train to Florence
3 Florence
Train to Salerno if you want to stay ON the Amalfi coast- then ferry to whichever town you have chosen
OR
Train to Naples then on to Sorrento for 3 or 4 nights . From Sorrento you can visit Pompeii, Amalfi, Naples, Capri easily.
3 or 4 Amalfi or Sorrento
Train back to Rome- put all your Rome nights at end avoiding a split stay
3 or 4 Rome
Fly home from Rome
If you decide to go to anywhere in Sorrento/Amalfi area you will need to make that decision quickly as lodgings book up early, you might not find much availability at all.
Since you only mention a possible day trip to Amalfi-
(which is not a good idea at all- there really is no good way to "do the Amalfi Coast from Rome"
then I'd say stick with the usual first trip and see the big 3- Venice, Florence, Rome
Fly to Rome-train to Florence
3 nights Florence
Train to Venice
3 nights Venice
Train back to Rome- put all your Rome nights at end avoiding a split stay
4 nights Rome
Fly home from Rome
train info here- both companies run fast trains and also have easy to use apps
https://www.trenitalia.com/content/tcom/en.html
https://www.italotreno.it/en