We are traveling on the Sun Princess out of Barcelona in May 2025....
Are you trying to do this as a day trip while the ship is in port in either La Spezia or Genoa? If so, I'd definitely choose La Spezia as it's a gateway city to the CT and much closer than Genoa...which is about 90 minutes by train from Genova Piazza Principe Station (closest to the port) to Monterosso, the most northernly of the CT villages. From La Spezia Centrale to Riomaggiore - the most southerly of the CT villages - is only 8 minutes by train.
IMHO, the villages themselves will be very crowded, regardless of how you get there or from where.
Editing to add: I don't think you need a tour either. It is important, however, that you make it back to the ship on time if indeed you're doing this from a port and have a boarding deadline. If uncomfortable with figuring out the trains (and maybe one ferry ride) plus keeping all 9 of you together, then I'd book a tour. We've only done the CT independently over multiple days so unfortunately I've no first-hand experience with tours.
I'm guessing this is the tour from La Spezia you're looking at?
https://www.shoreemotion.com/en/tour-details/1286/cinque-terre-tour-with-limoncino-from-la-spezia
Do be aware that the tour wouldn't be conducted by Shore Emotion. It appears to be a 3rd-party booking agency, and at quick glance I'm not coming up with much for reviews.