I spent 3 days in Porto Venere last September and while it as beautiful as advertised - if I posted my pictures you'd be sold - it is small and I doubt you could fill an entire day amusing yourself exploring there. We visiting CT more than a decade ago - before the tickets and permits and all the current stuff - and on the advice of everyone we left that memory in place and decided to explore other places. We took the ferry across the bay to Lerici and explored that side one day and I would go back to stay in Lerici, but honestly not in Porto Venere.
If you're interested you could catch the bus that runs along the coast pretty close to your cruise ship landing and travel to the southern side of the bay and explore there. You could also check the ferry schedule and ferry to PV explore it and then ferry over to the Lerici side before travelling by bus or taxi back to the cruise ship for the full "Bay of Poets" experience. Check out photos of Lerici, Tellaro and Sarzana's old town and see if that interests you.
For more Tuscan flavor Lucca about an 1:15 from La Spezia - usually with a change in tiny Viareggio. I like Lucca (and have spent several nice days there) but there aren't a ton of "sights". There are many, many churches and you can tour some of the big ones but other than the charming old town, ancient buildings, modern stores, many restaurants and cafes and Italian life going on.
You could also do the train from La Spezia to Pisa (1:15 1 change), tour Pisa and then Pisa to Lucca (:30) and see both before training back to LS. I think you would need to leave Lucca no later than 5:30/17:30 to get back to LS and then to the cruise ship by 8:00/20:00
I wish I knew more about the Ligurian coastal towns north of CT but I haven't explored there a bunch. I'm sure people here have strong opinions. I know Ligurian locals love Chiavari, we ran into an English couple on the train who had spent a week in Sestri Levante and we planning on coming back for a month but they reported hating Rapallo which is reportedly beautiful. But everything gets ritzier as you approach Portofino so I guess the level of rich and famous you're interested in dictates how far up the coast you go. The train station at that end is in Santa Margherita Ligure which is Portofiino adjacent and looks like a good mix of ritzy but large enough to explore. The train to Santa Margherita Ligure is an 1 to 1:30 hours depending on which train you take.
Keep in mind the cruise terminal is more than a mile across the city away from the train station so any train usage needs to be counted as time in what probably a taxi or bus ride away. I don't know if there is any cruise transport provided around the city like to the train station. If you plan on taking the train download the Trenitalia app for your phone to check schedules, buy tickets and track trains.
Have a great day ashore,
=Tod