We stayed in Porto Venere for several days and visited around the Gulf of Poets area some and I have a coworker who lived in San Terenzo for several years. You don't mention whether you will have a car but all of these places will require a bus to reach from La Spezia. There is also an occasional ferry that runs between Porto Venere and Lerici from the CT ferry area. A taxi from La Spezia to PV is a posted but crazy fee of 40e (a few years ago) while the but is about 15 minutes from the city center.
My pictures of PV will make you want to book a trip there but ultimately I found it way too small for the CT ferry traffic that regularly landed there. When the ferry stopped and crowds of tours people spilled out into the little shorefront area it was simply too much. Away from the port frontage are some charming streets and restaurants but it was overwhelmed.
Lerici has a marina and is the more 'resorty' and upscale area on the southern coast. There are some great little shops and places to eat although all the seafront places seemed questionable for the price, but this is pretty standard in beach areas. San Terenzo maintains some of it's quiet ex-fishing village vibe and feels calm and small compared to the more popular towns and it is connected to Lerici by a 1.5k long walkable seafront promenade passing all the little coves and blue beaches. Lerici has a restored castle to explore that contained an art exhibit while we were there and is connected to the footpath only neighborhoods on that hillside.
Tellaro is reachable by bus from Lerici is kind of a CT town before the tourists, but keep in mind that also means fewer places to eat or shops/services geared toward tourists. I didn't get to Sarzana but I've heard good things - especially about shopping and eating in the old city center - and the old section looks charming. It is about 15 minutes by bus from Lerici over the hill.
We had limited exposure to La Spezia itself and it is something of an 'everyday Italian town' there is a largely pedestrian shopping zone that, while not fancy, I found somewhat charming. I'm not here to champion La Spezia but I think it might be worth a bit more time than people rushing to other places give it credit for and it is certainly more charming that modern built Italian towns I've visited.
I'm really interested in visiting some of the towns on the coast - especially Chiavari, Moneglia, and Sestri Levante. I heard one that more local say they felt Chiavari was the perfect Ligurian town. Coming on the train we met and English couple who had stayed in Sestri for a few days and were in the process of planning to come back for 2 months the following year.
AS I said I'd love to explore this area - especially with the time to hike between the various towns which is completely doable but always require more time than I have.
My $.02, enjoy,
=Tod