Trying to set a transportation (train) budget

This totally does not answer your question regarding train fares but to me that itinerary moves way too fast. I’d cut down on the moves which will also cut train fares. I’d cut Sorrento, Naples, Pompeii, Amalfi Coast. Add nights to Rome, Florence and Venice for the same total number of nights.

When I check train fares I go to the train company in question and plug in cities. I buy tickets as far out as I can if there is a price break for doing that otherwise I buy them at the time.

Look up your routes on www.Trenitalia.com, and put the dates out one month - February 20.

Rome is Roma
Florence is Firenze
Venice is Venezia
Naples is Napoli

My adult daughter & I just did a 15-day Italy trip last September. I will add the link in case it helps with your planning. It was her first time and my 7th time there. Usually the larger touristy cities are more expensive both for lodging & food. For instance, our hotel in Ferrara was about half price compared to Rome.

Of the list you have shared, I usually advise to go to one water location if it’s just a 2-week trip. So, I would go to the Amalfi Coast or Cinque Terre, not both. And those take the longest by transportation, too.

https://community.ricksteves.com/travel-forum/trip-reports/mother-adult-daughter-italy-trip-aug-sept-2022

I think you first need to reduce the amount of travel, as you're moving around too much, spending too much time on trains. You have two nights per town. I'd suggest skipping the Sorrento / Amalfi area on this trip, and adding those days to Rome and Florence, perhaps.
I'd also suggest flying open jaw, to avoid backtracking.
Safe travels!

I agree with Pam. Rick's suggested itinerary has too much moving around, which leads to too much time packing up, sitting on trains and unpacking again--with too little sightseeing time. Every time you move you lose at least half a day, so when you figure six days for Rome, Florence and Venice, that's only about 1-1/2 days in each city.

Don't forget that the day you arrive in Europe, some of you (maybe all) will be significantly jetlagged and/or sleep-deprived. That first day may be nearly worthless for sightseeing. The day you fly home will also be mostly unproductive. Then consider all the half-days lost to switching hotels, and with that itinerary 13 days turns into more like nine real days.

Annabelle, I will also share the trip I did last June. Most of these locations would be cheaper than the Big 3 - Venice, Florence, Rome. If budget is keeping you from being able to go, some of these locations could help. And I had a fantastic time!

https://community.ricksteves.com/travel-forum/trip-reports/trip-report-please-do-not-comment-until-i-change-the-title-filling-in-the-report-thanks

@pam and @jean thanks for the info and link!
@pat and @acraven it looked full to me too but I just went with what Rick said to start. I can def cull stops. I am flying open jaw - arrive Venice and depart Rome. And I am not counting my arrival or departure days in my 15 touring days.
@jean thanks for the report. I’ll read it over. I have been to Italy several times. But never Rome or Amalfi coast. I took my now husband to Florence, Venice, Lake Como area many moons ago. Honestly I could forego another trip to Venice or Florence but it is on the boys’ list and I get that. It’s where I went my first time too. I’m happy to share the experience with them
Thanks everyone! I appreciate the input!

arrive Venice and depart Rome. And I am not counting my arrival or
departure days in my 15 touring days.

Is this 16 NIGHTS in Italy?
Perfect plan already— flying in to Venice, out of Rome.

But never Rome or Amalfi coast.

Oh!
You must go to Rome. Our favorite city.

The fewer location changes you make, the less your train/transpo cost will be.

Venice 3
Florence 3
Siena 2 or Orvieto 2
Sorrento 4- more to do in this region than CT
Rome 4

All easily done by train.
Off the top of my head- maybe 160-200 euros/person?

Get familiar with Trenitalia and ItaloTreno websites. Both have easy to use apps. You can check schedules and prices now. Book early for discounts if your plans are firm.

Choose lodgings that are walking distance from stations. Easy enough in all of the above.
Think about what sort of lodgings you will want. 2 hotel rooms/2 bathrooms? Apartment? Will you need to do laundry? Must have AC.
I think it’s best to stay in a hotel for a first entry to Italy. Hotel Abbazia in Venice is affordable and easy to get to, if available.
June is not so far off- you'll need to book lodgings soon.

16 yr old and 19 yr old

What are they interested in?

Trenitalia has discounts for people under 30 and a "me and you" promotion for two adults travelling together. Make sure you know if these apply to your proposed trips as the savings are up to 50%.

Maybe get the 16 and 19 year olds involved in planning the train trips.

this June
.

What is the best way to estimate our train budget

If you mean before June 11, Trenitalia has already uploaded on their official site all the top, walk-up fares called BASE.

You can't pay more than BASE, even if you click the "BUY" button 15 Minutes before the train departure. This would be your Worst-case-train-budget. Or the Cost-of-flexibility-budget, as you prefer.

On the other hand, if your plans are set you can pay a lot less purchasing discounted tickets.

Be aware that discounted tickets:

• They can't be purchased on the day of travel
• Changes are restricted and come at a cost
• No refunds
• they are capacity controlled, in short: discounted tickets can and will sell out

Italotreno is a private railway company that runs only high speed trains on major routes, they are cheaper than Trenitalia but they open their sales in 3 months-blocks. Italotreno's version of the BASE tickets is called Flex.

Also, take note that certain schedules are less expensive than others but also may involve multiple changes and/or longer travel times.