We are traveling throughout Italy this June/July. Should we get train tickets when our destination requires a train? Should we rent our cars in advance as well.
Tyrker's rule of thumb is good. The short distance trains that you buy on the spot are called "Regionalle".
- Regionalle = No discount for advance purchase, no reservations (get on and grab any free seat), tickets are valid on any train, but you must time-stamp them before use, to stop you re-using the ticket again later.
- Long distance trains (all other trains) = discounts for advance purchase. Tickets are only valid on the train listed on the ticket. Your reserved seat number is shown on the ticket.
For the full "Beginner's guide to Train travel in Italy*, see here: https://www.seat61.com/Italy-trains.htm
Times and "self-print" tickets on the official website: https://www.trenitalia.com/en.html
Trick - you need to use the real names (Italian names) of the stations, "Roma Termini", "Venezia S. Lucia" etc.
Car rental should definitely be arranged in advance, to get a better rate, to make sure a car is available, and if you need an automatic, to greatly increase the odds of getting one at all. Europeans drive stick shifts; if you need an automatic, you have to make sure the rental location has them (airports and major train stations in major cities are you best bet; small towns may not offer them).
But before that, what is your itinerary? We can then advise which places are best by train and which by car; where to pick up and drop off the car (if one is needed); and which train tickets will have discounts for advance purchase and which ones will not.
Please remember the cautions mentioned by Ken in your recent question about car rentals. Failure can cost big money.
Another reason to rent auto in advance -- not all companies have offices in every location. For instance, if you want to drop off a car in Orvieto to take the train into Rome, you're restricted to Hertz.