From Rome to Venice you can purchase fast train tix in advance for a nice discount- but they are non refundable so you need to be sure of your dates and time.
Same for Venice to Florence.
These tickets do not need to be validated as they are already date/time specific
The schedules will be updated June 12- so you can look now on a similar day to see what your options are.
Florence to CT is regional trains- no discounts- the price is the same no matter when you buy and they can't sell out. If you pre-purchase this ticket you do not need to validate- if you purchase at station on day of then you do need to validate.
CT to Rome is probably a combination of regional and fast train.
Use the actual train sites to buy your ticket
https://www.trenitalia.com/content/tcom/en.html
https://www.italotreno.it/en (fast trains on same routes only)
Both have very easy to use apps as well.
You will need to use the proper Italian name for stations
Roma Termini
Venezia S Lucia
Firenze SM Novella
CT- whichever town you are heading to
Read all about trains in Italy here:
https://www.seat61.com/Italy.htm