You didn't specify the exact destinations so I'm going to guess based on what you said.
Italy is usually crowded in the summer. Given your picture I suspect you may want to spend some time in Liguria. Usually, the trains are very crowded during weekends because people from close regions use to go there for a relaxing two days at the beach.
Depending on where you are leaving from you won't need to book any ticket in advance because there are regional trains. As a general rule of thumb (don't fully belive in this rule) trains named FRECCIA #COLOR (where #COLOR could be ROSSA / BIANCA / ARGENTO) require booking in advance. They are fancy trains and usually fast. For all other types you usually don't need to book in advance (if you don't mind a 2nd class seat).
In any case, TRENITALIA is where you have to look. From there, you can get all the information you need to know about a train (if you need to book or not, what type of coaches etc) and you can book your tickets from there (a lot of different types of payments are available).
For example, if you go from Torino to Camogli you can take a train with 2nd class non bookable (which means you are not allowed to book) coach only.
If you want to go to Venezia from, let's say, Milano you may have to book a seat on FRECCIA BIANCA. If you book it now you can have it for 19€ only. Since the economy seats are limited, the more you wait the more you are going to pay (if you manage to find a seat).
Trains in Italy aren't famous to be on time. Expect some delay, unfortunately they are common.