At the northern end of Vieux Nice lies Place Garibaldi dominated by the statue of the great man. The square was designed in the late 18th century by Antoine Spinelli, who also created the beautiful Chapelle du Saint Sépulcre. Close by is Muséeum d'Histoire Naturelle de Nice. Amongst the many exhibits is a remarkable collection of shells, some of them very rare, from all over the world - one of the best malacology collections in Europe.
Opposite the Natural History Museum is the Musée d'Art Moderne et d'Art Contemporain (MAMAC), an impressive array of square blocks and passageways, the work of Yves Bayard and Henri Vidal. The entrance is on a huge piazza and the collections include works by Yves Klein and artists of the Pop Art movement.
Heading east on the Promenade des Anglais you will reach the Rauba Capeu human sundial. The name means "steal hat" in the Nissard language and alludes to the fact that there can be a lot of wind on this promontory. Standing in the center of the sun dial you can see what time it is by looking at your shadow. Continue left for about 50m and you will see the bottom of the stairway to the Colline du Château. Climbing the steps you can visit the Bellanda Tower, built in 1830 and its small Maritime Museum. If you don't want to climb the many steps you can always pay to take the lift, just continue a bit further on Quay Rauba Capeu.
On top of the hill is a park with many beautiful trees, shrubs and flowers of the Mediterranean. Most rewarding are the extensive views over Nice and the Baie des Anges to the west and the Vieux Port and Mont-Boron to the east. Kids will relish the playgrounds and the chance to tobogganing. The château itself is long gone - it was destroyed in the reign of Louis XIV. The few ruins you see are of a medieval cathedral. Before you leave you might want to visit the Cimitière du Château, a Jewish cemetery, one of the most beautiful resting grounds in the world.
The Port of Nice, also referred to as the Vieux Port, is flanked by the Colline du Château to the west and Mont Boron to the east. Very picturesque with some fishing boats and a lot of expensive yachts. From here you can reach Corsica either with regular ferries (6-8hours) or the NGV high speed ferries in about 3 hours. The port is protected from the sea by a sea wall (dique) with a lighthouse (phare) at the very end - a good place to take a stroll.
Not far from the port, on 25 Boulevard Carnot, is the Musée de Paleontologie Humaine de Terra Amata, a little paleontology museum, which describes prehistoric fauna, flora, climate and tools. It is probably more for Nice's school children than foreign visitors.
The cathedral with its onion domes on Avenue Nicolas II (a side street of Boulevard du Tsarevitch) was built in 1912 with the "solicitude and generosity" of Czar Nicholas II, for the many rich Russians who vacationed or settled on the French Riviera. It is a real work of art inside and outside, beautiful! Recently the Russian Orthodox church tried to claim or as they say reclaim the property, read more about it here.
Cimiez, the former Roman Cemenelum, is one of the best residential districts of Nice. Many of the houses are in Belle Époque style with well tended gardens. Main sights are the beautiful Cimiez Gardens with the 17th century Cimiez Monastery with its Franciscan Museum, exhibiting many religious works of art, showing the the Franciscan way of life from the 13th century to the present day. The Cimiez Museum and Archaeological Site (entrance is on Avenue Montecroce) shows well-preserved Roman baths from the 2nd and 3rd centuries AD, vases, jewelery and other artifacts found in excavation here and around Nice.
The Musée Matisse (164 Avenue des Arènes de Cimiez) is housed in a beautiful red mansion near les Arènes, the Roman amphitheatre. Henri Matisse lived in Cimiez from 1917 until his death in 1954. Paintings, drawings and many of earliest works are exhibited here, mostly paintings but also some sculptures. Further south on Avenue Docteur Ménard, where the Cimiez district begins, is the Musée Chagall in a somewhat somber building. Chagall was born to a poor Jewish family in Russia in 1887. He moved to Paris and then to southern France. He died in St.Paul de Vence in 1985. Chagall's paintings portray scenes and events in a quasi child-like but very vivid way. He painted 17 huge canvases that form the basis of this museum, depicting scenes from the Books of Genesis and Exodus, and the Song of Songs.
A bit it out of the way, on 59 Avenue St.Barthélémy, is the small Museum of the Vieux Logis Priory or as it is officially known, the Musée des Arts et Mobilier de la Fin du Moyen-Age. Verify opening hours (very short) before you going there. The museum in this beautiful villa houses an interesting collection of religious art as well as everyday objects and furniture of the Middle Ages.