Do I need to reserve trains ahead of time?

If you are worried about finding a seat, then it is NO as trains rarely if every fill up. However, advance purchases on-line can have substantial discounts. BUT -- these tickets are basically no change, no refund. If willing to commit to a fixed schedule then purchasing in advance can be a very good deal. All trains, except Regionale trains, come with a mandatory seat reservation.

However, the one exception -- the Regionale trains - think bus on wheels -- are open seating. Cannot find a seat, stand till one becomes available at the next stop. The is no discount for these tickets so there is no point to early purchase. In fact on-line of Regionale train tickets come pre-validated with restrictions. Buy them in the station as needed. And then you validate (time stamp) just prior to boarding. Some of your trains will be via Regional trains.

To add to Frank's helpful post, this is a great intro to Italian trains:

https://www.seat61.com/Italy-trains.htm

You're not really "reserving", you're buying an actual ticket, unless you are using a Eurail Pass where you do have to buy a reservation for high speed Freccia trains and IC trains.

If you know you will be in Venice on such-and-such date, and you know you want to travel to Florence on that date, you can save money by purchasing a nonrefundable train specific ticket now. The ticket will only be useable on that particular train on that date, like an airline ticket. Example: for June 19, you could buy a ticket now for 24.90 EUR for the 9:35 train to Florence. If you waited to buy it at the station, it would cost 51 EUR.

Because most of the train connections between Florence and Monterosso are Regionali, you can wait to buy these in Florence. They are unreserved and always the same price. If it is after June 10, you can't buy them now anyway because the official schedule has not been finalized yet.

For Monterosso to Milan, on June 26, you could buy the nonrefundable ticket for the 10:35 IC train to Milan for 9.90 EUR that would cost 28.50 EUR on your departure day.

No experience with trains from Venice to Florence but good luck!

You have three levels of fares on all trains except Regionale. Super Econ, Economy, Base. The first two are the discount fares and you get them in either First class or Second class. The discount fares are limited in number per train and once sold out, they are gone. No last minute sales. Base is the standard, walk-up fare.

Thanks for all the responses. They were very helpful. I purchased the tickets from Venice to Florence.

One other question: Which is the correct train station for Monterosso? Monterosso or Monterosso Marche?

Monterosso, the other one is in Marche, southeast of Bologna.

Thank you Sam.

For Monterosso to Milan, on June 18th the fares are higher. I could only find a 57 Euro (2 people) leaving in the morning. Is it because it is still too early to buy tickets?

No, that is the base price. The Super Econ, and Economy tickets are sold out. (Or, it is possible that few if any discount tickets were available for the morning train.) That is the only fare available on that train at that time. The 5.17 am has 25 euro tickets for two. And 40 euro tickets for the 5pm train. All are intercity trains so you make lots of stops. There is not a high level of service out of Monterosso.