You need to validate only regional train tickets in Italy. If you have a reserved seat on one of the trains, it's not a regional train, so you don't need to validate that ticket. But it won't hurt if you validate it anyway if you aren't sure or are in a hurry or something.
Regional tickets bought from a ticket machine or a ticket agent (paper tickets) must be validated before you board the train. If you buy online for a specific day/time for a regional train, it does not need to be validated; you can even show it on your phone or tablet. It will have a QR code to be scanned by a conductor/ticket inspector, but you may not even see a conductor. I found the Trenit app on my phone handy for buying train tickets. No need to worry about validation, finding a ticket machine, or waiting in line to buy from an agent. And regional tickets cost the same whether you buy from the app or buy at the station, no matter when you buy them.
If you buy paper tickets from a machine or from an agent, you'll get a separate paper ticket for each train. Each ticket must be validated at the station where you board the train. You validate each ticket only once. If you validate it twice, the earlier time is the only one that matters.
You could validate all our tickets at Vernazza for that trip, IF you will board the last train within four hours of your original validation. For example, if you validate your tickets in Vernazza at 8:00, you'll be OK as long as you'll board the final train to Florence by 12:00. The trip to Florence isn't quite that long so you should be OK, but if there's a long delay and you miss a connection, you might miss the four hour window.
Vaporetto pass: swipe it at the dock to validate it each time you get on; you'll get a flashing green light from the validator box to show you're OK. Watch others do it. It's a little chip card, and you can even keep it in your wallet or your pocket or something if you can rub your pocket up against the validator box - as long as the green light flashes. The train tickets are old-fashioned paper tickets where the validation machine stamps the time when you slide it into the validation machine.