Travel from Venice a to Monterrosa

Use this website. It will tell you routes, times, train changes and transfer times.

https://www.bahn.com/en

The fastest route is via Firenze(Florence) and Pisa. It takes 5-6 hours. For changing trains I prefer to allow at least 10-15 minutes in Italy.

Just go to the Trenitalia site: https://www.trenitalia.com/en.html
and look up Venezia S. Lucia to Monterosso (note spelling, o on end), and enter your desired date and time.

This will automatically find the quickest route, using the recomended minimum time to change trains.
Where it says "changes", if you click on the little triangle it will show the times at interchange stations.
If you click on the 'i' beside each train number, it shows the stops for that train.

The ticket prices it quotes are for discount "self print" tickets. Ticket prices only ever go up, buy early to get the best pricesw.

Or the Italian national rail operator, Trenitalia.
https://www.trenitalia.com/en.html
Use Italian city names and stations. Venezia S. Lucia for Venice. Monterosso (not Monterosso Marche).
One issue is that regional trains beyond June 8 will not show up yet, so for planning purposes, use the same day of the week in the near future. schedules don't change much, if at all.

Fewest changes, but not necessarily the fastest or cheapest, will be a Freccia high speed train to Milano Centrale changing to an IC train direct to Monterosso.

Last year we took a Frecca train from Venice Santa Lucia to Milano Centrale. We took the 8:20 am train and arrived in Milan at 10:45 am. Then we took an Intercity at 12:10 pm that arrived in Monterroso at 3:05 pm. I opted for this route for the reason that we could purchase these tickets online ahead of our trip with rexerved seats on both trains, and because there was only one change of train. I discovered that because it is slower than going through Florence and La Spezia, trenitalia.com doesn't even list it. I bought the tickets on www.trainline.eu.

It doesn't list it, but there is no need to risk your money with re-sellers. Just make two separate purchases on trenitalia.com, leaving 15 minutes or more as transfer time at Milano Centrale.

You can buy in advance also the tickets via Florence, but it's true that there are no reserved seats on Regionale trains. If you miss a Regionale train for whatever reason, you can freely get on the next Regionale going in the same direction. If you miss the Intercity train from Milano Centrale to Monterosso you must always get a new ticket at the counter and it will be free only if you missed the 2nd train because of Trenitalia.

ls,

There are two main routes to get from Venice to Monterosso, one via Milan and one via Florence. I tend to prefer the Florence route. That trip will require a longer travel day, as you're crossing from one coast to another.

The solution I'd probably use is a departure from Venezia Santa Lucia at 09:35, arriving Monterosso at 14:53 (time 5H:18M, 2 changes in Florence and Pisa). If you buy tickets locally, that particular solution uses two express trains (Freccia, Intercity) which have compulsory reservations (provided with the tickets), and one Regionale train. You MUST validate the ticket for the Regionale train prior to boarding the train on the day of travel, or you may face hefty fines which will be collected on the spot.

If you pre-purchase tickets online that include a segment on a regional train, the regional train portion of the tickets will come pre-validated. If you buy tickets at a ticket window or kiosk, then you'll recive paper tickets good for any regional train on that segment that day, and those must be validated before boarding the regional train.

Thank you all for the great suggestions. I did book online with the trenitalia website. It looked to be much cheaper than waiting until we arrive there. We will go through Florence and Pisa and on to Monterroso.