Hi Suzanne
La Spezia you have a couple of options, Pisa, Florence or Lucca.
We have been to Florence a number of times andPisa too, but thought Pisa would be more interesting for our dd. We got the train - very easy to do, there is quite a walk to the station (20mins), but you could also get the little white train that goes from where the tenders come in (5 euros) to the station in about the same time you can walk.
Florence is a change on the train and I personally didn't want to risk it, so would suggest doing the Florence on your own, so
DCL take you there. It is a beautiful city and easy to walk around (although busy), with a good guide book.
Pisa you can walk through the town from the station (10 mins) via the shops which is interesting

Although there isn't a lot in Pisa, apart from the tower and the baptistry, a sort of church (do book your tickets beforehand, because they come in time slots and are limited). Lucca is a small medieval walled town, very pretty. We didn't do this, but someone else did Pisa and Lucca, which you could do if you are out early.
La Spezia is also a good shopping area and opens just as you are coming back, so worth allowing time to look around the shops.
Corsica is a lovely island, but you need to get out of Ajjaccio really to appreciate it. However, there is a good market (lots of fruit, veg, cheeses, wine and honey), just a stroll from the ship in the morning. There are a number of monuments to Napolean and some shopping areas which are interesting.
We did an open top bus ride along the coast, then back to the town late morning, when the shops shut for lunch, we walked to the little beach just out of town. There are some nicer beaches further along and you can get a bus No. 5 to these, which run there and back every 20 mins. Or a cab for 15 euros or so. Some of the beaches have pedalos, with little slides on them. The water is so clear and ideal for snorkeling.
To be honest, after so many busy stops, it was great to just chill out a bit.
Europe really is an easy place to navigate, we always do our own thing, either by cab or train and never have any problems.
Hope this helps.
Kate