Nine days from Nice to Genoa: the Riviera keeps going after France stops

Drop the bags and walk into Vieux Nice, which is four or five streets deep and completely unlike the boulevards behind it. The Cours Saleya runs along its southern edge and holds a flower and produce market every morning except Monday, when it turns over to antiques. Get there before ten for the flowers and the socca vans.

Then climb. The Colline du Chateau is not a castle any more, because Louis XIV's army demolished the fortress in 1706 and nothing much was rebuilt, but the hill it stood on gives you the whole Baie des Anges, the terracotta roofs of the old town on one side and the port on the other. There is a lift up from the eastern end of the seafront if the stairs look unappealing after a flight.

Come down for dinner in the lanes. Socca is the thing to start with: chickpea flour, olive oil and water, baked on a copper disc in a wood oven, scraped off and served in a paper cone with black pepper. It costs almost nothing and it is best from a place with a queue.

The hill of Cimiez, a bus ride above the centre, was the Roman town of Cemenelum before Nice existed as anything much. The baths and the amphitheatre are still in the ground, open and free to walk through, and the museum beside them explains what a provincial capital of the Maritime Alps actually looked like.

In the middle of the same olive grove stands a red seventeenth century villa holding the Matisse museum. He arrived in Nice in 1917 for his health, stayed for the rest of his life, and is buried in the cemetery a few hundred metres up the hill. The collection runs from the early dark still lifes to the cut-outs, and it is arranged so you can see the whole arc in an hour.

Downhill, ten minutes by bus or thirty on foot, is the Chagall museum, which was purpose-built in 1973 for one thing: the seventeen canvases of his Biblical Message, hung in the order and the rooms he chose. He came to the opening. There are also stained glass windows the size of a wall in the concert room, and a mosaic reflected in a pool outside.

Twenty five minutes west by train is a walled Greek foundation with a market under an iron roof that has run six mornings a week for well over a century, and a rampart you can walk along above the sea. The Marche Provencal fills the Cours Massena every morning but Monday with the produce of the hinterland: chard for the tarts, small violet artichokes, Menton lemons, and enough cheese to be a problem.

The old town behind it is compact and worth losing an hour in. Picasso spent the autumn of 1946 working in the castle on the ramparts and left the results behind when he went; the museum there is the reason most people come.

In the afternoon go out to the cape. The coastal footpath around it runs about two hours between rocks and pines, past the walls of some of the most expensive houses in Europe, and it is a public right of way the whole distance. Take water and shoes with grip. It is not difficult, but it is not a promenade either.

The train east from Nice runs on a shelf above the water through Villefranche, Beaulieu, Eze and Monaco, and takes thirty five minutes. Sit on the right. Menton at the end of it is a different temperature: the Alps come down to within a couple of kilometres of the sea and shut out the north wind, and the town has been trading on that fact since the doctors started sending patients here in the 1860s.

The old town stacks up a single hill in ochre and pink, and the way through it is a set of ramps and covered stairs called the Rampes Saint-Michel. At the top is a square paved in black and white pebbles laid in the arms of the Grimaldi family, and the church of Saint-Michel-Archange behind it, finished in the 1670s and still the biggest baroque building on this coast. A sixth of the names on the war memorial outside died in Italian uniform, which tells you where the border used to be.

In the late afternoon walk east into Garavan, the quarter that runs to the Italian frontier. It is one long slope of gardens: an olive park, an exotic garden of succulents, and the villa where a Spanish novelist built himself a Valencian fantasy in tiles and pergolas. Several open to the public on set afternoons, so check which one is open on the day.

Jean Cocteau spent his last years working in Menton and left more of himself here than anywhere else. In 1957 the town let him loose on the wedding hall of the town hall, and he covered it: a fisherman and his bride on one wall, the Saracen wedding on another, panther skins on the floor, and lamps he designed himself. Couples still get married in it, so it closes for ceremonies and opens on weekday mornings.

Fifteen minutes west along the seafront is the Palais Carnoles, which was the summer residence of the princes of Monaco in the eighteenth century and is now the town's fine art museum, free to enter, with primitives, a few Old Masters and a modern collection. The garden behind it holds one of the largest citrus collections in Europe, a hundred varieties of lemon, orange, grapefruit and things in between, and you can walk it without paying.

Finish above the town at the old chateau cemetery, laid out on the terraces where the castle used to be. It is where the winter colony ended up: English, Russian, German and Dutch names by the hundred, a Russian Orthodox chapel, and the grave of William Webb Ellis, the schoolboy credited with inventing rugby, who died here in 1872. The view along the coast from the top terrace runs to Italy.

Ventimiglia is ten minutes away and is where the trip changes country, currency habits and lunch times without changing coastline. Change trains there for Genoa and settle in for the run through Sanremo, Imperia and Savona, with the sea on the right and tunnels through every headland.

Genoa arrives suddenly and it is not pretty from the train. Get out at Piazza Principe, walk five minutes uphill, and the city rearranges itself. Via Garibaldi was cut in the 1550s as a street for the richest families in the republic to build on, and they built competitively: a dozen palaces in three hundred metres, most of them still with their frescoed halls and their courtyards open to the street. It is the finest single street of its kind anywhere and there is no fee to walk it.

For the first evening go up. A public lift climbs from the streets below to the Spianata Castelletto, a terrace on the ridge with the whole old town, the port and the cranes laid out underneath. Go at dusk, when the port lights come on and the sea turns the colour of the sky.

Two of the palaces on Via Garibaldi are museums and share a ticket. Palazzo Bianco holds Flemish and Dutch painting the Genoese bankers bought when they were lending to half of Europe; Palazzo Rosso keeps its own family's collection in its own frescoed rooms, and a lift takes you out onto the roof, which is the single best view of the old town there is. Give both of them a morning.

Lunch at the Mercato Orientale, a covered market from 1899 built inside the cloister of an old convent, where the produce hall is ringed by counters doing fried fish, focaccia and pasta to order. It is loud, cheap and entirely local.

In the afternoon go down into the lanes to San Lorenzo. The cathedral is banded black and white in the Ligurian manner, with a French Gothic front and lions on the steps that everybody sits on. Inside, near the south wall, stands a British naval shell fired from a warship in February 1941 that came through the roof and never went off. They kept it, put it on a stand and wrote a label.

The monumental cemetery at Staglieno opened in 1851 and is one of the great sights of Italy, which is not a sentence anybody expects about a graveyard. Genoa's merchant families commissioned life-size marble statues of themselves and their dead, in the clothes they actually wore, and the results run for hectares along colonnaded galleries. The most famous is a nut seller who saved for years to pay for her own monument and is carved holding her tray of nuts and doughnuts. Mazzini is buried here, and so is Oscar Wilde's wife. Go in the morning; it shuts in the early afternoon.

Come back for a long lunch in the old town, then spend the rest of the day at the eastern end of the city. Boccadasse is a fishing village of stacked pastel houses around a pebble cove, swallowed by Genoa a century ago and still entirely itself. Boats are pulled up on the stones, people swim off them, and the gelato place on the corner has a queue from four o'clock.

The walk there along the seafront promenade takes about forty minutes from the end of the bus route and is flat the whole way, with the sea on one side and the Genoese out running on the other.

One last Genoese idea before the train. In the sixteenth and seventeenth centuries the republic had no royal palace to lodge visiting popes, kings and ambassadors in, so it drew up official lists of private houses grand enough to do the job, sorted them by rank of guest, and assigned them by lot. Getting a cardinal was an honour; getting an entire imperial retinue was a bill. Forty two of those houses survive and the system put them on the UNESCO list in 2006.

Several stand on the two streets you have already walked, above the station and behind it, and their courtyards, staircases and entrance halls are open to anybody who walks in during office hours, because most of them are now banks, university faculties or council offices. Half an hour of pushing open doors is the best free hour in the city.