The answer depends totally on your, your spirit of adventure, travel smarts, and budget--plus what you want to do with your day in a fabulous part of France. As an example, you could rent a car for the day for about $100 - $120 total. Or you could take a full day tour which would probably cost you nearly $900 (for 5). Either way, you could see the same things. Another choice is to use public transit which might cost you around $50 - $150 total, depending on where you go. As to the "port city," Nice is very easy to explore on your own....especially if you enjoy walking. But many in this port want to go other places such as Monte Carlo, Eze, St Paul de Vence, Antibes, Cannes, etc etc. One issue you will need to face is that with a family of 5 on a Med cruise, you could easily spend many thousands of dollars taking overpriced tours in each port. My advice would be for your entire family to take a look at the NCL tour offerings, spend some time in a good book store such as Borders looking at guide books, and do some research on the internet. Than, once you have decided how you want to spend your days, you will be in a better position to judge whether you need a tour or can go off on your own. Every one of your ports can be explored on your own, or with a tour. The countries you are visiting are not third world countries, and they all have some pretty decent public transit, roads, etc.
Let me give you a good example. Lets say you decide that you want to explore Rome on your own. If you book the NCL tour, your cost would be $600 for the round trip bus. If you went on your own, you could take the free shuttle bus to the port entrance, walk 4 blocks to the train station, and take the train into Rome for less than $100 round trip (that is a total for all 5 of you).
Another example would be the Monoco and Nice tour. The cost for 5 would be $425 for the tour. On the other hand, you could walk from the tender pier (I assume you tender into Villefranche) and take the local train or bus to Nice. After walking around Nice you get back on the same train line and go to Monaco where you walk around and perhaps use a local bus. Your total cost would be less than $150. That's the way it is throughout your cruise.