Hello ashley,
If you are in or near Lucca with a car you could drive the 45 minutes or so to Sarzana and park at the train station and catch the 15-20 minute ride into La Spezia train station and switch the local train (or the ferry) if you can't get straight through to Monterosso on the train that works for you.
If you have a car I would also consider the south side of the La Spezia bay as an alternative to CT. It is not quite as scenic but between San Terenzo and the long waterfront walk to the more resorty Lerici and finally Tellaro which is like an undiscovered CT town because there is no train on this side of the coast. Even the non-coastal Sarzana itself has a reputation for good restaurants and some funky shopping in the old district although I haven't explored the town myself.
In June I'm sure Lerici and the other beach towns will be super busy but the tourism on the the south side of the bay tends to be Italians taking to the beach for the summer versus American tourists coming for the day. Given parking concerns at the beach in June maybe parking in Sarzana and taking the 15 minute bus over the hill into Lerici probably isn't a bad plan.
Just some alternate ideas if you're looking for some scenic Ligurian coast visits during your stay.
Have a great trip!
=Tod