Eze-Village (name is important here) and Monaco can be seen on same day e.g. bus 82 from Nice to Eze-Village around 9AM then same bus towards Monaco and Monte-Carlo for the afternoon. Then return to Nice by train from Monte-carlo train station. This way you'll see some of the coast, including Villefranche-sur-Mer (that you could visit from Nice by rifing on bus 81)
Bus no 400 for Cannes is 1 euro single fare. Train is more expensive. If you visit some of Nice, I'd skip Cannes and replace it by Antibes in the AM to see the market. Or go to St-Paul-de-Vence by bus, almost a full day trip.
I still suggest you keep a full day for Nice. Here is an itinerary I used to suggest for a few days in Nice (was initially intended for a week stay):
Of course you will have to help yourself with a guidebook. Since you will be there for almost a week, I suggest you get one that gives info info mostly on Nice and not Provence and Côte d’Azur. I have one by Hachette Books titled “A weekend in Nice”, it includes also a city map.
Websites like the Nice city’s Tourist Bureau website http://www.nicetourisme.com/GB/somtxt.html and one for the Old Town http://www.oldnice.com/ may be helpful.
There are musts like walking on the Promenade des Anglais , both looking at the sea, the beach and the hotels.
The Old Town and the Cours Saléya: Start from Place Masséna and walk towards the Promenade on Rue de l’Opéra. Make a left on Rue St-Francois-de-Paule, check the boutiques but don’t buy there (touristy prices way too high). Walk straight ahead to the Marché aux Fleurs (Cours Saléya). There is a market every day except Mondays. Walk to the end of the market and you get to the Chapelle du St-Suaire. Make a left on J” Gilly street . It ends on Rue de la Préfecture. Make a left there and walk until you reach Rue Ste-Réparate. Make a right and you will see the Cathedral Ste-Réparate and get on Place Rossetti. Very typical (look at the warmth of colours if you are there 1hr or 2 before sunset). A left on Rue Rossetti until you reach Rue Droite. Make a left on Rue Droite and walk to its end. Stop by Palais Lascaris (free visit) and some nice artist shops (check Christian Dury’s gallerie => he puts typical Nice things in Plexiglas frames). Rue Droite falls into Rue St-François then in Rue Pairolière. Both typical Old Nice streets with locals shops, food store, fish and meat stores, etc. Open your eyes and your ears and take a deep breath. Now you see the real thing. You get to the Place Garibaldi. Time for lunch. Help yourself, there are plenty nice spots at cheap prices.
From Place Garibaldi keep your right and stay on Rue Catherine Ségurane from where you will get a road that brings you up the Castle Hill (Colline du Chateau). Walk up to its top and have a look at the whole area (the Port, the beaches, the town). Stop by the Cemetery on your way down and wak to the Port. Look at the small fishermen boats on Quai des Deux Emmanuel. And walk back to town circling the Castle Hill by walking on the sea side to reach back Promenade des Anglais.
That one should take not far from a day long, especially if you stop often at boutiques or for an ice cream cone…
Another nice day is going to Villa Éphrussi Rotschild by Bus 81 (starts from Gare Routière). It accepts regular bus tickets (4 euro for a day, around 15 euro for a week pass with unlimited travel). Ask the driver for the name of the bus stop right at the doorsteps of the Villa. A superb mansion with 7 fantastic gardens. 2 hours of pure delight. Wanna have fun and see the real thing once again. When you get out of the Villa walk to your left. The road goes up a bit, walk on the road to your left. It goes down towards St-Jean-Cap Ferrat where you can stop for lunch and get back to Nice with Bus 81 once again. Or you could even walk on the seaside towards Bealieu-sur-Mer using the trail called Sentier des Douaniers. It is about an hour walk and you could then visit Beaulieu, its Villa Kerylos and get back to Nice by train.
You are there on a Friday? Take the train to Vintimille (Ventimiglia in Italian), a small Italian town where you have a well-known market. You can’t get lost. When you get out of the train station walk straight ahead. 200 ft and it’s crowded, you just found the market! Before getting in it, look on your right for the food market. You wanna see Italians wheeling and dealing? Get in there and have fun? Back to the real market, you can get good deals on leather, cashmere wool, etc. Beware the Africans dealers. It’s no good deal!. The morning is enough for the market. There are some nice Italian restaurants at cheap prices if you don’t mind walking a block or 2 on your right (eastbound), getting away from the market. Back on the train to Nice, get down at Monaco (not the Ste-Dévote Chapel exit at the back of the train but the one at the head of the train). Have fun for the afternoon in Monte-Carlo.
Already 3 days gone. Like art? Get a car or take the bus to Vence and St-Paul-de-Vence. It’s about ¾ hour from Nice and take the day for both villages. If you rented a car and feel like shopping with the locals crowd, stop by St-Laurent-du-Var Cap 2000 shopping centre. A huge one.
Museums? Like Modern Art? The Mamac Museum of Modern Art is a must. Then you have both Chagall and Matisse Museum on bus route 15 from the City centre.
Wanna shop? Rue Jean Médecin, rue de l’Hotel des POstes. Rue Gioffredi are all street you should not miss and you have the Centre de l’Étoile shopping Centre on Rue Jean Médecin along with the Galeries Lafayette, the Monoprix for food and clothes and the FNAC for records and books (along with Virgin Megastore across the street). There are walking only streets close to that area and they are fuil of shops and restaurants.
You could take the bus to Eze-Village, a medieval village that must be seen. Menton along the coast is also quite nice to see and has its own Old Town.
Westbound towards Cannes, you have Antibes and Juan les Pins that can be reached easily by bus.