Ten days in Austria: the cities get smaller and the mountains get bigger

Vienna's cathedral is not on a square of its own. You come round a corner in the shopping streets and it is simply there, sooty and enormous, with a patterned tile roof and two towers that do not match because building in Gothic stopped in 1511 and the north one never got finished. The nave is free; the guided route to the red marble tomb of Emperor Frederick III is not, and is the part worth paying for.

Then sit down, because the coffee house is the fastest way to understand this city and the first afternoon is the time to learn it. You are buying a chair, not a drink: one coffee, a newspaper on a wooden stick, and as long as you like. Hawelka is five minutes from the cathedral and the least polished of the famous ones, dark and cramped with paintings stacked up the walls.

In the evening, the opera. Standing places are sold on the day of the performance for a few euros and the queue for them builds about two hours ahead, so eat early. You get a rail to lean on, a small screen with the words in English, and a better view than several classes of seat.

The Hofburg is a district rather than a palace, six centuries of building with no agreed style. Most visitors queue for the imperial apartments and never find the treasury, which is the better half by a distance: twenty rooms of Habsburg crowns, robes, reliquaries and the regalia of the Holy Roman Empire, in the oldest corner of the complex and usually quiet.

Cross the Ring after lunch to Maria-Theresien-Platz. The picture collection is administratively part of the same museum and ten minutes away on foot, and it holds Bruegel by the roomful, the largest gathering of his paintings anywhere, plus Vermeer, Caravaggio, Titian and Raphael and a floor of Egyptian and Roman rooms below. The building is worth the ticket on its own: a marble staircase, ceiling panels by a young Klimt, and a café under the dome.

Finish with coffee at Central, in the courtyard of a nineteenth century palace with vaulting overhead and a pianist most afternoons. It is the touristy one and it is still worth an hour, for the room rather than the food.

Take the U4 out to Schönbrunn first thing and be at the gate when it opens. The Habsburg summer palace is the most visited building in the country, a kilometre of yellow frontage, and two tours are sold: twenty two rooms on the shorter one, including Franz Joseph's narrow iron bed and Empress Elisabeth's apartments, and forty on the longer, which adds Maria Theresa's interiors and is the better of the two. No photographs, no backpacks. It has been on the UNESCO list since 1996.

The gardens behind are free and much larger than they look. Walk up the slope to the colonnade for the view back down over the palace and the city, which takes about twenty minutes and loses most of the crowd.

The same U4 line runs back to Karlsplatz and the other Vienna, the one that could not stand the emperors. The Secession went up in 1897 as an exhibition hall for artists who had walked out of the academy, a white block topped by a sphere of gilded laurel that the city immediately nicknamed the golden cabbage; Klimt's Beethoven Frieze runs round a room in the basement. Across the square stands a baroque church with two carved columns bolted to the front of it like a Roman forum, which sounds absurd and looks magnificent at dusk.

The Belvedere is two baroque palaces facing each other down a terraced garden, built for Prince Eugene of Savoy on what was then open ground outside the walls. The upper of the two houses the Austrian national collection, and almost everybody is there for one gold-leafed square metre of it. The Kiss is smaller than the posters and permanently three deep, so go at opening. The rest of the Klimt room deserves more attention than it gets, and the treaty that ended the Allied occupation and gave the country back to itself was signed upstairs in 1955.

Walk north into the Landstrasse district for an hour of the twentieth century. Hundertwasser's council block from the 1980s has uneven floors, trees growing out of the windows and not a straight line in it, and people still live there, so you look from the street.

Finish across the canal in the Prater, an old imperial hunting ground that became a public park and then a permanently shabby funfair. The Ferris wheel at the entrance went up in 1897, burned in 1944 and was rebuilt with half its original cabins, and is the one Orson Welles rides in The Third Man. Every other great wheel of that generation, in Chicago, London and Paris, was scrapped long ago. Fifteen gondolas, twenty minutes, and the best evening view in Vienna.

Take a mid-morning train and you will be in Salzburg for lunch. The line runs west along the Danube through St Pölten and Linz, flat and unremarkable for the first hour, then lifts into the lake country and the first proper mountains come up on the left in the last half hour. Salzburg's station is on the north bank, fifteen minutes' walk from the old town.

Spend the afternoon on the main lane. Getreidegasse is narrow, arcaded, and hung end to end with wrought iron shop signs showing what each business sells, a survival from when most people could not read; even the fast food chains have had to commission their own. It takes ten minutes to walk and most of an afternoon to enjoy.

At number nine is a tall yellow house where the Mozart family rented the third floor from 1747. Wolfgang was born there on 27 January 1756 and lived in it until he was seventeen. Inside are his childhood violin, a clavichord, portraits and a great many letters, and it is better than the souvenir shop downstairs would lead you to expect.

Go up first thing. The Hohensalzburg has held its ridge since 1077, was extended by one archbishop after another, and no besieging army ever took it. A funicular saves a steep twenty minute walk and is the oldest in Austria. Inside are state rooms, an armoury and a marionette collection; what you actually come for is the rampart, with the Alps on one side, the town flat below on the other, and a single house standing alone in a field, which belonged to the executioner because nobody would live next to him.

Come down the far side into the courtyards of St Peter's abbey. The graveyard is a small dense plot of wrought iron grave markers under the cliff, still in use, and cut into the rock above it are rough chapels reached by steps, a couple of euros and twenty minutes.

Finish across the square at the cathedral, where the comparison with Italy stops being a figure of speech: white, vast, consecrated in 1628 to a design by a Lombard architect, with three bronze doors and a dome that lifts the whole square. Mozart was baptised at the font just inside and later worked here as court organist. The excavation museum below shows Roman floors and the foundations of the Romanesque church that stood here first.

Twenty minutes south of town on bus 25 is the summer palace of an archbishop with an unusual sense of humour. Hellbrunn was built in the 1610s and its garden is rigged: hidden jets in the paths, in the grottoes, and most famously in the stone dining table, where the host stayed dry and every guest did not. The system is original, water-powered, and still operated by a guide who will get you. Go on a warm day and expect to be wet. The park around it is enormous, free and full of picnicking families.

Back in town, cross to the north bank for Mirabell. An archbishop built the palace in 1606 for the woman he lived with and their children, a scandal that eventually cost him his job, and the gardens in front are laid out so that the fortress sits precisely in the gap at the end of the central axis. Look for the marble grotesques along the west wall and the gilded hall upstairs, free when nothing is booked into it.

Then climb the hill on the western edge of the old town for the evening. Augustinian monks have brewed here since 1621. You take a stone mug from the shelf, rinse it in a stone trough, hand it over to be filled from a wooden barrel, and carry it out into a chestnut garden looking over the roofs. Food comes from a corridor of market stalls, or you bring your own, which people do.

The Salzburg to Innsbruck run is the best two hours of the trip. The line climbs out of the lake country, threads the Tyrolean valleys with rock walls close on both sides, and ends a quarter of an hour on foot from the arcades. Sit on the left. Reserve a seat in summer, because the trains are short.

Innsbruck's historic centre is a handful of arcaded lanes and its emblem sits at the end of the main one: a late Gothic balcony jutting over the square under a roof of gilded copper tiles, put up around 1500 to mark Maximilian I's marriage to Bianca Maria Sforza and to give the couple a box seat for the tournaments below. It is smaller than the photographs suggest and the painted houses around it are half the pleasure.

Two minutes away is the thing most visitors miss entirely. The court church was built to hold Maximilian's tomb, and around the sarcophagus stands a double row of bronze figures at twice life size, ancestors real and invented, blackened by four centuries and known here as the black men. King Arthur is among them, cast in Nuremberg and one of the finest bronzes of the Renaissance anywhere. The emperor is not actually in it: he was buried at Wiener Neustadt and the monument was finished long after his death. Then climb the old watchman's tower on the square for the first proper look at the wall of rock you are going up tomorrow.

Go early, before the valley haze builds. The funicular starts underground beside the court gardens, surfaces on a bridge over the river and reaches Hungerburg in eight minutes; the stations are Zaha Hadid's, curved white shells like drifted snow, built to replace a line that had run since 1906. Two cable cars carry on from there, and about half an hour after leaving the old town you are standing at the top station just under the summit ridge.

The last stretch is a gravel path, ten to fifteen minutes up to the top of the Hafelekar at a little over two thousand three hundred metres. The main chain of the Alps stands across the south, the valley is a thin grey line directly below, and the city looks close enough to drop a coin onto. Come down as far as the middle station for lunch, which has the same view and a wall to sit behind when the wind gets up.

In the afternoon go the other way, out to the wooded hill southeast of town. Archduke Ferdinand II built Ambras in 1563 and lived there until he died in 1595, filling it with the most famous cabinet of curiosities in Europe: armour, coral, clockwork, portraits of people with rare conditions. It survives in its original rooms, which almost nothing of its kind does, and the claim that it is the oldest museum in the world is a serious one. Upstairs the Spanish Hall has an inlaid wooden ceiling and twenty seven full length portraits of Tyrolean rulers along the walls.

The Imperial Palace is the third of the great Habsburg residences and by a long way the quietest. Building began around 1460 and Maria Theresa gave it the baroque face it wears now between 1754 and 1773, so most of what you walk through is hers: white and gold rooms, family portraits by the hundred, and the apartment kept for Empress Elisabeth. Ninety minutes covers it and there is no queue.

Then the train, or the bus to the airport, or the two hour run north to Munich if that is where your flight home is. Whichever it is, the station is fifteen minutes' walk downhill from the palace door and there are lockers on the concourse.

Ten days ends where the road over the Alps has always gone, which is a reasonable place to stop or an obvious place to keep going. Verona is four hours south through the Brenner and one of the great railway journeys in Europe, and the tickets cost almost nothing.