There are two types of train in Italy:
1) Long distance or high speed trains, under a number of brand names.
2) "Regionale" or local trains.
For long distance or high speed trains, when you buy a ticket it is for a specific train (like air tickets are) It will say train (date time, train number) and a seat number on the ticket. The ticket is only valid for that seat on that train. These tickets are cheaper the earlier you buy.
For Regionale trains, you do not and cannot reserve seats. A ticket is valid on any train, and you need to timestamp ("validate") the ticket just before you get on the train, to stop you using the ticket twice. Thre is no discount for advance purchase of Regionale train tickets.
The only time you ever need to buy a resevation as a separate purchase is when you have a rail pass, and passes are very rarely a good buy or Italy.
Venice - Milan and Milan - Florence are high speed trains (1).
Florence - Assisi and Assisi - Rome are by Regionale trains.
See here for lots more info: http://www.seat61.com/Italy-trains.htm
You can look up train times, see what sort of trains they are and buy tickets at the Italian Railways site: http://www.trenitalia.com/tcom-en