In early June 2012 we (on the Magic) stopped at La Spezia. Most people headed to Florence, but we took a train to Cinque Terre. It's a very easy DIY trip, but kids would probably be bored as its more about the scenery, quaint fishing villages, and overall ambiance. Rick Steves' guide book explains how to get there and what to see and do very well.
The train station is a bit far from the pier so we took a taxi. There is a special train ticket that allows you to get on/off the train as much as you'd like, and some versions include the shuttle boats as well. We just did the train and stopped at 3 of the villages. Didn't have time for all 5, but then we were in no rush, just enjoyed each village until we were ready to leave.
We spent a very pleasant 90 minutes over lunch, which is actually my favorite memory of the day. We have a local chef from La Spezia and he suggested some dishes we should try. Most of the restaurants had a chalk board menu posted and when we just chose one that had the dishes we wanted to try. I don't think we'd have been disappointed at any of the restaurants.
DW wanted to transfer between villages using the boat, but they don't operate is the sea is rough, and as they are small boats it can ba a long wait to board one, so we opted not to use them. Turned out to be a good choice as there were very long lines for the boats in at least two of the villages.