Three days in Graz: warm, young, and left off everybody's Austria trip

Begin with the strangest building in Austria. For two centuries Styria was the front line against Ottoman raiding, and rather than keep a standing army the province kept a warehouse: four floors of weapons and armour ready to be handed to whoever turned up. It was sealed and preserved when the threat receded, and it is still there, some thirty thousand pieces racked exactly as they were, the largest arsenal of its kind left anywhere. None of it is decorative. These are working pikes, muskets, breastplates and helmets, made in bulk, dented and repaired.

Take the guided tour or you will get very little from it: without a guide the rooms are rank after rank of steel with no labels at all. English tours run at set times, so ask when you arrive. The upper floors get punishing in July.

At six, be on Glockenspielplatz. A gable opens in a house on the square and two carved wooden figures in Styrian costume turn and dance to a bell tune, which takes about three minutes, draws a crowd of thirty and is exactly as daft as it sounds. Then eat: the old town is full of small places doing local food for a third of Vienna prices.

The hill in the middle of Graz is the reason the city exists and it is now a park. There are three ways up from the square at its foot: a funicular, a lift bored straight through the rock, and two hundred and sixty steps zigzagging up the cliff, which is the one to take if your knees allow, because the wooded paths and the terraces on the way are most of the pleasure. Allow an hour and a half to wander it.

The fortress that stood on top was demolished by Napoleon's troops in 1809 after the city paid a ransom to spare two buildings. One of them is the clock tower, which has been the emblem of Graz ever since and has a peculiarity worth knowing before you look up: the big hand tells the hours and the small hand the minutes, the reverse of every other clock, because it originally had only the one hand and the second was added later. From the terrace beside it the whole old town lies below with its Renaissance courtyards open to view.

Come down and cross the river. In 2003 Graz was European Capital of Culture and spent the money on two things that should not work and both do. The first is a gallery in the shape of a bulbous blue creature with light nozzles on its back, dropped among sixteenth century roofs, known to everybody here as the friendly alien. Inside it is a straightforward contemporary exhibition space, and the ramp up to the top floor with the town visible through the skin is better than most of what gets hung in it.

Start on the water. The other thing built for 2003 is an artificial island moored in the middle of the Mur: a twisted steel shell holding a small amphitheatre, a café and a playground, joined to both banks by footbridges. It was meant to last a summer, the city liked it, and it stayed. Ten minutes and a coffee, and the river running under the glass is the point.

Then take tram 1 west for fifteen minutes to Eggenberg, which is the reason to give Graz a third day. The palace was built from 1625 as a model of the universe in stone, and once you know the scheme you cannot stop counting: four corner towers for the seasons, twenty four state rooms on the main floor for the hours, fifty two windows in them for the weeks, and three hundred and sixty five windows in the outer walls for the days. The state rooms run in a single enfilade with painted ceilings, inlaid floors and tiled stoves, and they are only open on a guided tour.

The park is free to walk if you have a tour ticket and full of peacocks that have been there for generations. Trams run back to the centre and on towards the station, so the palace works as the last thing before a train.