Hello Go Blue, and welcome to the forum!
You'll want to use the official site of Trenitalia which is the state train company of Italy: https://www.trenitalia.com/en.html
You'll want to use the Italian city names for best results when searching on train sites: Roma for Rome, Firenze for Florence etc. They also have an app for use in country that also let's you track trains in realtime and validate tickets etc.
Fast trains - Frecciarossa - between big cities and these come with assigned seats on a specific train and tickets can be purchased early at a (sometimes sizable) discount but come with restrictions. The earlier you buy the cheaper the tickets but you're trading economy for flexibility. Tickets for fast trains are generally available two months ahead.
Local trains called regionale tickets are always the same price - just buy them when you need them. Also if you have a reserved train and you miss it because of the train being late and it's the train companies fault they are required to re-book your ticket even if it is a restricted change ticket. Of course if you just miss a train with a restricted ticket the restrictions apply.
If you book from Bologna to Arezzo you get both tickets - potentially a fast train from Bologna to Florence and then a regional train from Florence to Arezzo. Twenty minutes is plenty of time to change in Florence as all the tracks are laid out in a row. Bologna train station, however, is built on several levels like an airport so head to your train without delay because it may be further than you think.
The man in seat 61 is a really good basic primer on European train travel: https://www.seat61.com/train-travel-in-italy.htm
Have a great trip,
=Tod