I would not consider taking the train from Monterosso to Siena. Way too many train changes, and if you have luggage it’s not fun.
La Spezia is the closest city to Monterosso with rental car offices. Both Hertz and Europcar have locations near the cruise port. Coming from Monterosso you would take a train to La Spezia Centrale and from there a taxi to the rental office of your choice at the port (less than 2 miles away).
You can also rent the car at Pisa. Pisa is further south, so there would be an extra 40-60 min on the train from La Spezia (and extra train fare). Like La Spezia, at Pisa no rental office is at the station. There are offices downtown within a short taxi ride, or at the airport which is connected to the station via an automatic people mover (6 min ride). Be aware that picking up a rental car at the has extra charges/fees.
If you pick up in La Spezia the fastest way to Siena is Via Viareggio, Lucca, Prato, Florence using the A12, A11, A1 (exit Firenze Impruneta) then the Raccordo Firenze Siena Superstrada (a toll free divided freeway).
If you pickup at Pisa, it is a bit longer route. Basically you take the SCG FIPILI (a divided superstrada) to Firenze Scandicci, then the A1 for a short distance (to Firenze Impruneta), then the Raccordo as above.
The shoreline route via Cecina and Volterra, which I’ve experienced many times as a kid spending beach time in Cecina, would not be my choice, unless you dislike someone in your party and you want them to get car sick and throw up along the way. If you want to pass by Volterra on the way to Siena, it’s easier to drive the Val d’Era (Era Valley) from Ponsacco. This route (via Ponsacco and Volterra) would be slower than the ones describe above, since it’s mostly on a regular two lane highway after Ponsacco, but probably more scenic.
None of the routes should have major traffic outside of commute time. You might encounter traffic around Florence if you get there during the evening commute.