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We did a cruise on Ocean Village (1) and did the organised excursion to Monte Carlo. The guide offered to take anyone who was interested into the Casino for a guided tour. The entry fee to the Casino isn't included in the excursion fee and we paid a few Euros extra to get in. You have to leave your bags and camera at reception for security reasons. There is a dress code. Men should wear long trousers/shirt, ladies wear dresses/skirts. They do look at your footwear and don't let in people in trainers/flip flops, so something smarter is required. There was an Edith Piaf exhibition in one room, showing her costumes and jewellery which was interesting. I'm glad I have seen it but it was out of my price range to play at the tables!
I have done Monte Carlo several times. We tendered in Villefranche - I would recommend hopping on the local train. It's only a few euros and takes you into the heart of Monte Carlo. It gives you the freedom to see the sights and is only about 30minutes each way. A taxi ride from there or even Nice is quite hairy. The roads are narrow and very windy.
You CAN go up Versuvious without a tour, but it's not within walking distance of the ship. You'd have to take a taxi to the mid-way drop-off point, then hike the rest. you may have a problem getting transport back from there. You're best bet is to purchase a basic excursion to get you there and back on that one.
Why not do your research on the Internet about places you will be visiting. I usually do and have not gone on any organised excursions or years. Public transport in most places is far superior to ours and you can always get yourself around, even taxis arre much cheaper but degotiate with the driver first about the fare otherwise you leave yourselves open to being ripped off. Have funn
There is a great bus service in the south of france around Nice/Monaco etc.At most of the ports you can just walk into the town and there is plenty to do and see. Do not forget to see the perfume factory at Grasse and you can catch a bus there from Nice. Get the timetable on line.