Hello Mark&Mel,
As you’ve heard, the villages of the Cinque Terre are indeed better visited by train rather than by car. There is very limited parking at each and the roads are narrow and steep. So my recommendation would be to head to La Spezia and leave your car in the parking lot near the rail station (you should find free spots but check in case you end up on a paid area) and from here take the train to either Monterosso (most northern town) or Riomaggiore (the first town you reach). Connections to the Cinque Terre are short and run throughout the day. You can check the
Train Schedules ahead of time to better plan your day.
Since you will have a car, you can also make your way down to the coast to Massa and down to Viareggio, where you can enjoy both the sea and beaches as well as shopping, nightlife, pubs and discos, especially in Viareggio, Pietrasanta and Forte dei Marmi which become during the summer a really fashionable destination for Italian families and young people.
If you are seeking beautiful sea and you like rocks I definitely advice you to head to Livorno, which in addition to be a characteristic harbor town offers also beautiful landscapes and high rocks from which you can improvise sky diving, if you can !

Here you can park along the street, and walk down to the coast through natural paths which try out your deftness but show you gorgeous sights overlooking the blue water of the Tyrrhenian sea.
Especially advised is the sea at Calafuria, Livorno.