Let me try to answer your question with another question. Why do you want a taxi? Keep in mind that taxis in France and Italy are pretty expensive...especially when you factor in the very weak dollar. I cannot give you a good answer regarding La Spezia since we have never been there as a cruise stop. We have traveled in that area with our rental car (when on extended land trips). I think the place that jumps out in the immediate region is Cinque Terre which are actually 5 villages connected by a great hiking trail. I will tell you that it take the better part of a day to hike the entire trail and its quite strenuous in places. Given our experience in Italy I would expect there to be some taxis available at the pier, but they are going to try and negotiate with those who want to hire them for the day (the going rate used to be about 300 Euros per day). Another option would be the train which does connect all 5 villages. They used to run hourly which make it possible to hike between 2 or more of the villages and use the train to get to some of the other towns. If you plan to go to Cinque Terre by taxi you will have a problem since two of the towns cannot even be accessed by road.
As to Nice (yes, Nice is nice) I am curious as to where your ship will dock or tender. Most tender into Villafranche although there is a small docking area in the Southern end of Nice that can be used by smaller vessels (we actually docked there years ago on the Marco Polo). Keep in mind that all the towns along the Riviera are connected by a single train line that is pretty much like a commuter line. Its possible to use that train to get almost anywhere from Monte Carlo to Cannes. If you are planning to go to Eze and/or St Paul de Vence this is a little more complicated since neither of these towns in right on the train line. France is one of the few places in Europe where the taxis do not necessarily cue at the dock, so if you truly want to hire a taxi for the day you should consider arranging for a car/driver on-line.
Now, you ask should you go with organized excursions. My DW and I have never, ever taken an organized excursion in the South of France (and never will) but we are the exception. In France we generally rent a car or, if its convenient, we might use a train or even a local bus. However, many will argue that for a first-time visitor an organized tour is the best way to cover the maximum sites in the shortest time. You will find some pretty good info on various guides if you use google (you will probably find a lot of references to posts on cruisecritic).
I will often recommend that cruisers take a good look at the various tour options offered by the ship and get an idea of the major sites in any region. Than, you can get an idea of what you want to do and you can do some homework to determine if its practical to do it on your own.