Train from Venice to Cinque Terre

The quickest route is: VENEZIA SANTA LUCIA - FIRENZE SMN - PISA - LA SPEZIA - MONTEROSSO.
This route requires at least 2 changes at the following stations:
FIRENZE SMN (Florence)
LA SPEZIA or PISA
However many options might require a change both at PISA and LA SPEZIA (for a total of 3 changes en route)
You don't need reservations for anything, you can buy on the spot when you depart from Venice, however FRECCIA trains and INTERECITY (IC) trains will have reserved seats, while REGIONALE trains you can seat on any available seat (like on a subway in NYC).
If you buy FRECCIA or IC trains in advance you might be able to get a discounted fare, however those come with strings attached (no refunds or changes if you miss the train you book).
You need about 10 min to make changes in Florence and even less in smaller stations like Pisa or Spezia.

The other route is via Milan:
VENEZIA SANTA LUCIA - MILANO CENTRALE - MONTEROSSO
This route takes longer, but potentially is the only one that might require only 1 change (in Milan)
However there are several options also requiring a change in GENOVA (Genoa) or other towns like Sestri or Levanto.

www.trenitalia.com
Enter from VENEZIA SANTA LUCIA to MONTEROSSO
Select time and date, but if your date of travel is far in the future, enter a date in the next week. Timetables don't change much from year to year, or season to season.

Just to level set - keep in mind because of the complicated infrastructure on each end I would expect you to be spending 7-8+ hours door to door from Venice to any CT town. You are crossing Italy at almost it's widest point from coast to coast.

Plan accordingly,
=Tod