What is the quickest and best way to get from Monterosso al Mare to Venice?
Meredith,
The best way to get from Monterosso to Venice is via train (IMO), and the quickest trip is a departure at 09:24, arriving 14:20 (time 4H:56M, 3 changes at La Spezia, Pisa and Florence). There are numerous other trains on that route with fewer changes but slightly longer travel time. Note that your destination station will be Venezia Santa Lucia.
You can view all the solutions on the Trenitalia website.
You may decide that a slightly longer travel time works for you, in order to make just one connection, which is an option via Milan. Do plan to pack light to make these train connections simpler, as well as because you'll navigate stairs and cobblestones en-route to your hotels in Monterosso and Venice.