Nine days in northern Italy: What you see when you skip Rome and Florence

Start at the top. The cathedral is worth twenty minutes inside and an hour on the roof, and the roof is the point: you walk on the stonework itself, between the flying buttresses, at eye level with a forest of carved spires, with the Alps visible on a clear winter day.

It took from 1386 until well into the twentieth century to finish, in pink-white marble brought down a canal dug specially to carry it, and no other Italian city built anything like it.

The glass-roofed arcade off the north side of the square leads to the opera house, which from outside looks like a bank. That was deliberate: a plain front, so that the attention went to what happened inside.

Your slot is your slot: be at the door twenty minutes early, because they will not hold it. You are dehumidified in an airlock, let into the old refectory in a group and given fifteen minutes.

It is a ruin and it is still overwhelming. Leonardo painted it on dry plaster rather than wet so that he could keep revising, which meant it began flaking in his own lifetime, and the wall has since been bombed, whitewashed and restored over twenty-one years. What survives is the composition, and the composition changed European painting.

Ten minutes away is a plain grey box of a church whose entire interior was painted by Bernardino Luini and his workshop, with a wall dividing the public half from the nuns' half and both sides frescoed. It is free and usually empty, and it is the best thing in Milan that nobody has heard of.

South of the centre is the part of the city that predates everything else. Sixteen Roman columns stand in a row in front of a basilica founded in the fourth century, when Milan was an imperial capital and Rome was not.

Then back to the middle for the twentieth century, which this city did better than anywhere else in Italy: Futurism, which was invented here, and the design and fashion industries that followed it.

In the evening go down to the canals. Milan had a whole network of them, built to bring the cathedral's marble in and goods out, with locks that Leonardo worked on. Most were paved over in the 1930s and two survive, lined with bars and full of people from about seven.

An hour and a quarter east across the plain, and you walk out of Verona's station into a town about a fifth the size of the one you left.

The first thing you meet is a Roman amphitheatre standing in the middle of a large paved square with cafes around it. It is the third largest to survive and by far the most used: gladiators, then a quarry, then bear-baiting, and since 1913 an opera house that puts fifteen thousand people on the stone steps on a summer night.

North of it is the old Roman forum, still the market square, with painted house fronts around it and a fountain in the middle carrying a Roman statue with a medieval head. Sit under the arches at six with a glass of Valpolicella.

Start with the walk west along the river that most visitors never make. Twenty minutes brings you to a Romanesque basilica with a striped front, a rose window and forty-eight bronze panels on the doors, cast in the twelfth century and cut with a directness that makes the later art in the city look overworked. Mantegna's altarpiece is inside, where he put it.

Walk back along the water to the red brick castle the medieval lords built to keep their own citizens out. Its museum is famous among architects: an Italian architect stripped the fake medieval restoration off it in the 1960s and left the joins showing, so the Roman wall, the castle and the modern intervention read as three separate things.

In the afternoon cross to the far bank and climb. The Renaissance garden on the slope has cypresses, a grotto and a stone mask cut into the rock face, and from the top of it a path runs up to the old fort with the whole city below.

An hour and ten minutes east and the train crosses a causeway onto the lagoon, which is the point at which people start looking out of the window. Santa Lucia opens onto the Grand Canal and the first thing in front of you is water.

Drop the bags and walk to Rialto, which is where the city began. The bridge was finished in stone in 1591 after several wooden predecessors fell down or burned, and it was the only crossing of the Grand Canal until the nineteenth century. The shops built into the span are original to the design.

Then eat standing up. The lanes behind the market are thick with small bars serving cicchetti, and two or three of them in a row is dinner.

Everything today sits within four hundred metres, which sounds easy and is the trap: this is the one square where all of the city's visitors are at once. Take the earliest basilica slot you can get. The mosaics are lit by daylight rather than floodlights, so they change through the morning and are best while the sun is still low.

The palace next door was a government building, a court and a prison at the same time, which is a strange sequence to walk through. The Secret Itineraries tour goes into the administrative rooms and the cells behind the public route and is the better version if the standard one sounds like a lot of gilt.

Then get on the water. The island opposite is one stop across the basin and it solves a problem Venice creates: from inside the city you can never see it. From the bell tower there, you can.

A north to south walk across the city, which is the best way to understand its shape. Start in Cannaregio, where the word ghetto comes from: the Jewish population was confined here in 1516 on the site of an old iron foundry, and because the island could not grow sideways the tenements are the tallest buildings in Venice.

The afternoon is two buildings a hundred metres apart in San Polo, and between them they hold more painting than most national galleries, all of it still where it was put. The Frari has Titian over the high altar and Titian's own tomb down the side aisle. Behind its apse, a confraternity hall that Tintoretto spent more than twenty years covering with fifty-odd enormous canvases.

That is the argument for this half of the city. Almost everything famous in Venice was made for a specific wall in a specific light, and here you can still stand where the painter expected you to stand.

The last morning belongs on the walk back towards the station. There is a small church in Cannaregio faced entirely in coloured marble, freestanding, with barely any room around it, built in the 1480s to house a painting of the Virgin that people believed had performed miracles. It looks like a jewellery box that has been left in the street.

After that, buy something in a bakery and take the waterbus down the Grand Canal rather than walking. It costs the price of a ticket and it is the cheapest sightseeing in the city.

Marco Polo airport is a bus or a boat from Piazzale Roma, next to the station, and the trains west go from Santa Lucia.