Be sure to go to the church of St. Peter (1198), built on the site of an ancient church of the 5th century. Great place for meeting both sunsets and sunrises.
The territory near the church, including the viewpoint on the roof, is open around the clock.
Initially, you might think it's a castle, but not. The huge Doria Castle (XII-XIV centuries) is located on a mountain. Open to visitors.
The island of Palmaria is perfect for trekking. On the next photo it is to the left. It has a beach, two restaurants, mountains, forests and abandoned military installations. You can get there by boat from the Portovenere quay (from 8:15 to 18:30). Just be sure to find the schedule of the last boat!
Getting there: by bus 11/P from La Spezia, find here the schedule. On a boat from La Spezia or Cinque Terre towns, find the schedule on the website of the Consorzio Marittimo Turistico, which operates boats - http://www.navigazionegolfodeipoeti.it/en/imbarchi-orari-tariffe.html. If you are by car, then park on Via Olivo, from this location N 44 03.110 E 9 50.191 and further towards La Spezia.