Transportation in Italy???

To learn about the trains: https://www.seat61.com/train-travel-in-italy.htm

Trenitalia and Italotreno actually operate the trains. Omio, Trainline, and Rail Europe are travel agencies and don't always show all the trains, only the ones they can make a commission on. Nonetheless, the travel agencies may have less stringent credit card purchasing requirements when buying online. Trenitalia and Italo have apps that can make purchases easier.

Orvieto, Florence, Venice, and Verona are very walkable, although Orvieto has a funicular to get from the train station to the hill town. In Venice, a vaporetto pass is handy to get around and really the only way to see the Grand Canal in full and get to outer islands like Murano and Burano.

Milan you can get a multi-day transit pass to use the Metro, trams, and buses.

Paper ticket: need validation for entry and scan for exit.

No, just validate before boarding regional trains with no need (or way) to scan at exit. Italo trains and Trenitalia Freccia and Intercity train tickets are specific to that train and day and have seat reservations, so no need to validate.

The website for the national railroad is Trenitalia.com, and there’s also a privately- owned company, italotreno.com. Italo has limited, high-speed routes, whereas Trenitalia has various levels of trains and is far more comprehensive in where it goes.
Orvieto: across the street from the train station, you can catch a funicular that goes up the cliff to the historic center. It’s very walkable once you’re up there, but you could take a bus if you prefer.
Venice: there’s a system of water buses (vaporetti) that go on the large canals, around the island, and to some of the islands in the lagoon. You can buy 1-, 2-, 3,-, or 7-day vaporetto passes. Venice is also very walkable.

I was just in Italy a couple weeks ago and used the trenitalia app. On the regional trains it asked to validate on the app before boarding. It was a simple press of the button on your phone app.

Italotreno now does sell tickets that combine one of their high speed trains with a regional train, so they can book you between a lot more places now.

Trainline is a good option for buying tickets, as they show you both Italo and Trenitalia options. Also trainline does allow you to use English place names that may be more familiar to you.

Trenitalia for fast and regional trains
Get the app

Rome is very walkable
Public transportation can’t get you close to most sites anyway as they are in pedestrianized areas
Suggest a taxi for an early am Vatican tour

Orvieto
Take the funicular/bus combo to center from station
Bus drops you at Duomo
Walk everywhere else in town

Florence so compact. Train station is close to historic center
Walk everywhere

Venice walk and vaporetto
Get a length of stay vaporetto pass since individual tickets are expensive
Be sure to validate every time before boarding

Verona
You might want a taxi from station to center/lodging depending on where you are staying
Otherwise walk everywhere-center is pedestrianized

Milan
Taxi to center/lodging from station
Sites in center are all walkable
Last Supper is a longer walk from center

Otherwise there are trams/buses and metro

The early am Vatican tour meet at

https://www.google.com/maps/place/Antico+Caff%C3%A8+Candia/@41.9068085,12.4497463,15.64z/data=!4m6!3m5!1s0x132f61a088cf47b9:0xe2bf6c84839e08b2!8m2!3d41.9072955!4d12.4501059!16s%2Fg%2F11p5jd_wl8?entry=ttu

I think it is close to the metro station and should be able to use metro to arrive there. Am I right?

For taxi in Italy, I heard Uber is not available.

Some websites suggest itTaxi app. Is this right?

Summarize trains in Italy

use Trenitalia and get used to Italian city names

Since city to city trains I am going to use (Rome, Naples, Florence, Venice, Verona, Milan) are not regional trains, I felt that I can directly board the train after the purchase on app. No validation at entry or exit.

I don't know if Circumvesuviana Railway to Pompeii is regional train or not.

As to Pisa, Siena, I will take guided tour. No trains involved.

The entrance to the Vatican Museums is in front of this Coffee Bar
You can use the Metro to get there, if it's a convenient option from where you are coming from. The station you get off is called OTTAVIANO.

ItTaxi app is correct.
However ItTaxi and Uber have negotiated a deal whereby you can use the Uber app to summon a regular taxi via ItTaxi.

From Venice to Verona, being relatively close, you might decide to take the cheaper regionale trains. If you do so, you'll need to validate a paper ticket.

The circumvesuviana is not part of Trenitalia. It's a regional commuter train that serves the Naples metro area.

Station names:

Roma Termini
Napoli Centrale
Orvieto
Firenze SM Novella
Venezia Santa Lucia
Verona Porto Nuova
Milano Centrale

Oddly in Austria, Uber and taxis were the same cars. I suppose they are priced the same as well?

is The circumvesuviana ticket e-ticket or paper one? does it need validation for entry and exit? For paper ticket to Verona, validation for both entry and exit?

which app to buy The circumvesuviana ticket?

Circumvesuviana tickets cannot be purchased online
Only at the station
Validate before boarding

No “exit” validation at all
Not sure where you got that

Several Youtube videos said “exit” validation.

Is this weekly card (24 euros) worth buying in Rome? I need use metro to Vatican and Colosseum I think. But I don't know all other places.

https://www.atac.roma.it/en/tickets-and-passes/cis

All tickets are validated before boarding (like trains) or as soon as you board (buses/trams). Validate means inserting in a machine that affixes a date/time stamp. If you don't validate you could use the same ticket many times

Here is what I learn about
E-ticket: pre-validated so no need of validation for entry.
Paper ticket: need validation for entry

https://www.italiarail.com/booking-basics/what-does-it-mean-validate-rail-ticket-or-rail-pass

Best resource for trains in Italy
https://www.seat61.com/train-travel-in-italy.htm

Here is what I learn about E-ticket: pre-validated so no need of
validation for entry. Paper ticket: need validation for entry

https://www.italiarail.com/booking-basics/what-does-it-mean-validate-rail-ticket-or-rail-pass

that is out of date for e-tickets

that website is a travel agent - a reseller - not the train company. Maybe why it is out of date.

Rome: buy INTEGRATED WEEKLY CARD (where to buy? airport?)
https://community.ricksteves.com/travel-forum/italy/suggestions-on-rome-visit

Last I knew, you were only spending 3-4 days in Rome, and one of those days was going to be a day trip outside of the city. So there's no need to buy a weekly pass. I'm curious why that's something on your purchase list?

That is due to my experience in other European cities. There I would better to buy day pass instead of each ticket. May not be the case in Rome

Several Youtube videos said “exit” validation.

This Youtube video shows no such scan on exit on the Circumvesuviana to Pompei Scavi.
https://www.youtube.com/watch?v=_OFuE2o8znU

For Milan, I bought a 24 hour pass for ~8 euro that worked for us.

https://www.atm.it/en/ViaggiaConNoi/Biglietti/Pages/HowtogetaroundMilanbypublictransport.aspx

You don't need to buy a weekly pass for Rome if you're not spending a week in the city. A 72-hour ticket is also available:

https://www.atac.roma.it/en/tickets-and-passes/roma72h

That's 72 consecutive hours: once you validate it the first time, it expires 72 hours later. If you don't, for instance, validate it until noon on Tuesday, it wouldn't expire until noon on Friday so you can stretch it across 2 full days and parts of two others.

These sorts of tickets do not work for transport to/from Fiumicino airport.
Rome is a very walkable city so you will likely use transport less than you think you will. That said, explore other ticketing options on the ATAC site:
https://www.atac.roma.it/en/tickets-and-passes

For iOS there is a free app with a 5-star rating called Trenit! (I am not associated with the developer in any way, I just use it). Enter your start- and end-point and your chosen date/time, and it will search both Italo and Trenitalia (hi-speed and regional) so you can see all options and prices in one place. If you click the Buy button for any option, it launches the website (in Safari) of whichever company runs the train you chose so you can buy the ticket(s).

It doesn't track train status or loyalty programs or anything else--it's just for finding trains and positioning you to purchase tickets. For comparison, Italo's app is rated only 2 stars on the App Store--according to reviews, it will let you buy tickets but not choose your own seats. Trenitalia's app is rated 3 stars.