As noted, Sorrento is not actually on the Amalfi Coast, but it is an ideal place to stay if you are (as you said) using public transportation to make a variety of daytrips. It's not a particularly good place to have a car, even if your hotel has promised you a free parking space and daily in-outs, which is unlikely. Perhaps you mentioned Salerno because you found online that the main railroad agency runs trains there from Rome, without having to change in Naples. Have you used Google Maps to see how far Salerno is from Sorrento?
You did not give the month of the year, which makes a big difference in driving from Salerno to Sorrento. (Is that your scheme?)
As many have posted here in past threads (see Search box top center of desktop site), a typical way to get from Rome to Sorrento (getting to the Amalfi Coast is harder ... ) is a low-cost advance purchase Frecciarossa ticket to Naples, and the dirt-cheap Circumvesuviana from there to Sorrento. However, the Circumvesuviana can be so crowded and unpleasant that for four persons, I would consider a private transfer ... if your luggage will fit into a vehicle you can book for the trip. A related issue is whether you need a taxi anyway upon arrival in Sorrento, which could cost 1/4 of the private transfer cost, again, if your luggage will fit into one cab.
We took the Circumvesuviana to Sorrento, but paid for a private transfer back to Naples to avoid the fierce (month of May) crowds and musicians on the Circumvesuviana. We enjoyed the ease of daily daytrips without luggage on the Circumvesuviana, however.