" I know we will have to change trains in Milan."
Not necessarily. There are two main routes between Venice and Monterosso, one via Milan and one via Florence. Of the two the Florence route is often slightly faster (5.5-6 hours) while the Milan route may have fewer changes (~6.25 hours).
If I'm reading your post correctly, you plan on connecting with a train right after an international flight? If that's the case, many of us here don't recommend buying advance tickets, but rather just buy tickets at the station when you arrive. The reason for that is that many of the trains you'll be using come with compulsory seat reservations, which are specific to train, date and departure time. If you miss the train you've booked due to flight delays, long queues at Passport control or whatever, the tickets are worthless and you'll have to buy more at full price (Base fare). If you're caught on a train that has compulsory reservations with the wrong tickets, you'll likely face hefty fines which will be collected on the spot!
If you decide to buy tickets locally when you arrive, one other point to note is that if you'll be using Regionale trains on any part of your trip, you MUST validate those prior to boarding the train or again, hefty fines collected on the spot! Tickets from Venice to Monterosso could be as much as ~€70 / ~US$78 PP at Base fare (but will depend on which train is used).
The easiest solution is likely to take the bus from the airport to Venezia Mestre and connect with the train there. The other option is the bus to Piazzale Roma and Venezia Santa Lucia station (which is right on the Grand Canal) and connect with the train there. Mestre is a bit closer to your destination, but only very slightly.
What time does your flight arrive? Which route you'll be using to Monterosso will depend on which train you connect with, so you'll have to deal with the changes associated with that route. If you travel via Firenze S.M. Novella, you'll change there and also possibly at Pisa Centrale and La Spezia Centrale (you'll be approaching Monterosso from the south). If you travel via Milan you'll approach from the north via Genova and Levanto.
I agree with a previous reply that it would have been preferable to use an inbound flight to Milan, Pisa or Florence. Adding 6 hours or more to the trip after a trans-atlantic flight is going to mean a dreadfully long and tiring day.