Let's start from the Sea.
The Albert 1er Garden
This garden was given the name of the Belgium king who fought bravely against the germans during WWI.
It covers, like all the green ribbon a river, the Paillon.
You'll find here the Théatre de Verdure (green Theater) were live shows take place (theater, concerts...) and where, since 2011, the Jazz festival also takes place. In summertime, you can watch films here, under the stars.
There is a merry go round in the garden, too, and several artworks like the Arc monumental. (It is a huge piece of steel with the shape of an arc of a circle oriented towards the sky.)
Now and then it houses exhibitions about books, cooking and miscellaneous topics...
Afer the Jardin Albert 1er you'll get to the Massena square, which can be considered the center of the city.
La Promenade du Paillon
This large and newly created garden with adult trees bought and brought here at great expenses is located where the former bus station used to stand. There are plenty of playgrounds for the children. Some water fountains hidden in the ground will surprise the most absent-minded. Like the "Prom" (the real one), it has become one of the inhabitants' favorite place to have a walk and on week ends it is overcrowded.
La Promenade des Arts
If you go further, you'll find the Nice National Theater, then the MAMAC and eventually the public Library. Climb the stairs to access the terrace between the theater and the MAMAC and see the exhibited sculptures (Calder, etc...) (The stairs on the building flanks are steep. An easier access is located between the MAMAC towers, in front of the library entrance. The stairs there are not as steep.)
Beyond, on the library's roof there is a very pretty garden. To see it you have to walk around the library, to the Square Head. Unfortunately, this garden is usually closed, which is a pity.