What's a good 3-day itinerary for Innsbruck?

Three days, three altitude bands. Day 1 stays flat in the Altstadt, where the Hofkirche, the Hofburg, the Golden Roof, and the Dom zu St. Jakob all sit within 800 metres of each other. Start at Café Munding on Kiebachgasse around 8:30 for Topfenstrudel and a Melange. The Melange is thick, the Topfenstrudel warm, and the dining room smells like butter and old wood. By 9:30 the Golden Roof balcony on Herzog-Friedrich-Strasse catches full morning sun, and the tour groups have not arrived yet. Cross the courtyard to the Hofkirche by 10:15. Emperor Maximilian I commissioned this church in 1553, and the 28 bronze figures lining the nave weigh about 2 tonnes each. Your footsteps echo off the stone floor between them. The Hofburg next door takes about 45 minutes if you skip the audio guide. Lunch at Ottoburg, a 15th-century tower on Herzog-Friedrich-Strasse, runs 14 to 19 EUR for a Tiroler Gröstl. After lunch, the Dom zu St. Jakob stands 2 minutes south, rebuilt in 1717 with a Cranach Madonna above the high altar.

Day 2 goes vertical on the Nordkette. The Hungerburgbahn funicular departs every 15 minutes from Congress station, a 5-minute walk south of the Altstadt. Zaha Hadid designed the four stations, and you'll recognize the swooping steel canopies before you see any signage. From Hungerburg, switch to the Nordkettenbahn cable car. Seegrube station sits at 1,905 metres, and the air temperature drops roughly 10°C from the valley floor. Wind cuts through a T-shirt up here. If the sky is clear, continue to Hafelekar at 2,256 metres for views across the Karwendel range into Bavaria. Mind you, the round-trip costs about 39 EUR without the Innsbruck Card, 0 EUR with it. On the descent, stop at Alpenzoo Innsbruck near the Hungerburg station. It opened in 1962 as Europe's highest-elevation zoo, at 750 metres. The ibex enclosure and the cold-water aquarium housing Danube salmon are the two exhibits worth the most time. Dinner at Lichtblick on the 7th floor of Rathaus Galerien faces north, straight at the Nordkette ridgeline you walked that morning.

Day 3 heads south of the Inn River. Tram 1 from Museumstrasse reaches the Bergisel stop in 12 minutes. The ski jump, rebuilt by Zaha Hadid in 2002, has a panorama platform at 50 metres and a café with floor-to-ceiling glass. On a clear morning the Inn Valley stretches east toward Hall in Tirol and west toward the Arlberg. Walk 10 minutes downhill to the Wilten Basilica, a pink-and-white Rococo church that tends to be empty before noon. Lunch at Gasthof Bierstindl on Klostergasse costs 12 to 16 EUR for Kasspatzln, Tyrolean egg noodles baked with mountain cheese and topped with crispy fried onions. After lunch, take the Sightseer bus to Ambras Castle. The Renaissance grounds need about 90 minutes. The Spanish Hall's painted ceiling is the single room worth the full ticket price. Maria-Theresien-Strasse, the Altstadt's main boulevard, runs 400 metres south from the 1706 Annasäule column.

Buy the 48-hour Innsbruck Card for 53 EUR. It covers every cable car, museum, bus, and tram in the city zone, which makes Day 2 alone worth the price. Walking distances stay under 6 kilometres per day. Day 1 covers about 5 kilometres on flat ground. Day 2 adds cable-car elevation but only 4 kilometres of actual walking, mostly downhill from Hungerburg. Day 3 runs to 6 kilometres with one tram ride and one bus. Late June valley temperatures currently sit around 22°C, dropping to roughly 5°C at the Hafelekar summit.