Hello Joy,
As noted high speed trains come with seat reservations and specific times and are most expensive "day of" as an option. There is the potential that they can sell out but that depends on demand, time of year etc. The local regionale trains can't sell out and there is no reason to buy tickets ahead of time.
Checking the Trenitalia site: https://www.trenitalia.com/en.html
- Verona (Verona Porta Nuova) to Bolzano (Bolzano Bozen) - There are some fast trains but more local trains. It only saves you 45 minutes so it's up to you
- Bolzano to Bologna (Bologna Centrale) - There are a few direct fast trains that take 2h 32min while direct while the direct slow train takes 4h 4min. Many of these routes have changes so you would need to look at them and decide what time and how much hassle works for you.
-Parma to Varenna - It is possible to get a fast train from Parma to Milan and save you about an hour but Milan to Varenna is always the same (see below)
-Varenna (Varenna-Esino) to Milan - Local train run by Trenord. It technically has a first class sometimes but it is just a few seats - don't bother with that.
You have several routes with sections that are served by fast trains but not all of it. You essentially need to choose between time saved with a pre-booked scheduled train you have to catch and a less scheduled but slower train option. You're not on any really long hauls so the time savings - while nice - don't strike me as critical.
The ticket item in the schedule will show you what kind of train it is and how many changes. Click on the ticket in the purchasing app and it will show you all the details - type of train, times, length of time in the station for a change. If you buy a changeable fast ticket and you miss it you can use the ticket to re-book (paying any difference in cost) but if you buy a super saver ticket and miss it because you (not the train) were late you lose it.
Download the Trenitalia app and you can check schedules, track trains and buy tickets from your phone.
I know it's a lot but it's not as confusing as it seems. Come back if you have more questions.
Have a great trip,
=Tod