Nice is a great place to spend some time – we always stay in Nice and travel in to Monaco. The old town in Nice is lovely, shame you are not there in the morning as the flower market and the fruit/veg is set up in the mornings with most of the stalls starting to clear away early afternoon. I would suggest the bus tour is probably the best way of seeing Nice and you can get on and off at the various stops and pick up the next bus which comes past. This goes from the Promenade Des Anglais by the Albert 1 Gardens – depending where you are staying there may be another stop more convenient. You get a headset which tells you the history of the area, and takes you to all the places which may be of interest including the harbour, the Russian quarter, Roman ruins, various other places of interest. The bus just loops. The other suggestion is to take the lift up to the tower which overlooks Nice – the views are amazing. This can be accessed through a tunnel to get the lift or if you are feeling energetic you can walk up!
Re Monaco, the problem you will find is that due to the action on Saturday/Sunday you are limited as to what you can see/do in Monaco. You can go round the zoo and the car collection of Prince Rainier, and you should I think be able to walk up to the Palace, but how easy it will be to get round at this time, I am not sure. It is worth making the effort however, as you can go to the Cathedral and there are tours around the palace. The problem is on these 2 days, a lot of the F1 fans go up there to try and get views of the circuit.
Re restaurants in Nice, you are spoilt for choice. The market in the old town is full of restaurants, most of them are very good but get very busy. It really depends what you want to eat – there is an Italian influence in the area. Every year we have been there seem to be new restaurants appearing although we have a couple of favourites which we tend to go to more than others. It really is a case of what your preference is food wise and also your budget. There is something to suit everyone.