Although you'll be travelling next week, you may still be able to save some money with advance purchase tickets. If you'd rather wait to buy tickets, you can certainly do that but you'll be paying the highest price tier (Base fare).
You'll only be able to "hop on the next one" if the train is a Regionale. The express / high speed trains have compulsory seat reservations, so tickets for those are specific to train, date and departure time. If you "hop on" the wrong express train, you'll face hefty fines which will be collected on the spot.
If you decide to buy tickets locally, you also MUST remember to validate (time & date stamp) tickets for Regionale trains prior to boarding the train on the day of travel. Failure to do so will likely result in hefty fines!
The quickest and easiest option appears to be a departure from Venezia Santa Lucia at 09:35, arriving Vernazza at 15:13 (time 5H:38M, 2 changes in Florence and La Spezia). The first segment is via Freccia high speed train, which will have compulsory seat reservations. Your ticket will specify your Coach No. (Carrozza) and Seat No's. (Posti). The latter two segments are both Regionale, which don't require reservations.
If you want to buy advance tickets, have a look at www.trainline.eu or the Trenitalia website. The Trainline website is very user friendly once you're registered on the site, and they also have an App. They sell tickets for several of the rail networks in Italy, as well as rail networks in other parts of Europe (France, Germany, Spain, etc.) and prices are the same as charged by the rail networks.