Could you provide details on the trips you'll be taking from Firenze SMN to Riomaggiore, such as train numbers, departure times or whatever? It sounds like you've bought tickets for the specific Regionale trains you'll be using on that route, which of course have to be validated prior to use. I would have suggested buying those locally at a Kiosk, as that would have made the process easier.
I rarely buy tickets in advance, so I've never encountered this situation before, especially with Regionale tickets. I believe you'll have to use one of the Kiosks to print out paper tickets for the Regionale trips you'll be making, according to the instructions they provided. You'll likely be using one train from Florence to La Spezia and then another from there to Riomaggiore. That will be easier to understand once I've seen the train details mentioned above.
What I'd do in the same situation would be to print out the tickets for all Regionale trips at the Florence station the day prior to travel, according to the instructions. You'll notice on THIS website that there's an option on the machine to "Collect Your Ticket" (scroll down to see the appropriate photo), which is the option you'll be using. You could also have a look at THIS website. The machine will print out tickets that resemble a card.
On the day of travel, simply validate the tickets for each segment prior to boarding the train. That means validating the ticket for the Florence-La Spezia segment in Florence, and the ticket for the La Spezia-Riomaggiore segment in La Spezia. Keep in mind that Regionale tickets have a "shelf life" once validated so it wouldn't be a good idea to validate them all at the same time. In reality, your ticket may not be checked on the final segment to Riomaggiore, as the trip is so short (however I'd still recommend validating).
Regarding the change in La Spezia, the Validation machines should be on the platform or in the tunnels between platforms, so it shouldn't take long to validate them (the validation process only takes about 5 seconds per ticket). One point to mention is that the Validation machines may also be bright yellow. Have a look at THIS website for an example (scroll down about half way on the page).
As mentioned, I've never had to deal with this situation, but hopefully Roberto or one of the others will be able to add some further information.
One other point to mention is that if you buy tickets in the Cinque Terre for travel on the local trains, these may resemble an old fashioned theatre ticket (very small). As I recall, it's necessary to insert these into the validation machine on the left side, otherwise the machine won't "see" them.