I recommend not bothering with the car in Rome, the train ride to Salerno will be quick and fun.
Driving on the Amalfi Coast can get tricky but is certainly more convenient and flexible than the local buses - just be warned that during the weekend it's going to be very, very busy with traffic and parking both
Depending on your plans you could also get around locally by ferry:
https://www.travelmar.it/en/index