From San Gimignano or Poggibonsi (wherever you are staying) you could drive to Levanto or La Spezia and from either town take the train.
La Spezia is a bit closer from where you are coming from, therefore it would be about 10-15 min faster.
However, if you are going to the farthest towns from La Spezia, i.e. Monterosso or also Vernazza, then Levanto is probably better, because Monterosso is only 4 min train ride from Levanto (and Vernazza is right after Monterosso), therefore you could save a few minutes by driving a bit longer to Levanto and taking the train from there.
But if you are going to Riomaggiore or Manarola, which are the two easternmost villages, therefore closer to La Spezia, then parking in La Spezia would be faster, because it's a shorter drive and also a shorter train ride.
I wouldn't take the train all the way from Poggibonsi, because it takes 4 trains, therefore too many train changes (Poggibonsi to Empoli; Empoli to Pisa; Pisa to La Spezia: La Spezia to Cinque Terre village).
If you are going only for a night or two, then driving back to Tuscany, then you could keep the car. If you want to park your rental car, closer to where you are staying, instead of leaving it in a parking structure in La Spezia, then Monterosso offers the most parking options. Some hotels even have parking in the new part of town (Fegina).