Innsbruck Airport: Take IVB bus F, not a private airport transfer. It runs to the centre in roughly 15–20 minutes (about every 15 minutes; less often on Sundays). Budget roughly €3.30 for a standard city ticket; a taxi is only worth it with heavy luggage or an awkward late arrival.
Innsbruck Hauptbahnhof (main station): This is the right rail arrival point: direct trains connect Munich, Vienna, Salzburg, Zürich, Bolzano/Bozen and Venice. The Old Town is an easy 10–15-minute walk via Maria-Theresien-Straße; do not take a taxi just to reach the Golden Roof. The bus station is immediately beside the station for regional coaches and valley routes.
Munich Hauptbahnhof: For many international arrivals, Munich is the practical alternative gateway. Use a direct EC/Railjet or regional train to Innsbruck; the rail journey is roughly 1 hour 45 minutes to 2 hours, then Innsbruck Hauptbahnhof is walkable to the centre. Avoid planning around the Brenner motorway by car on holiday weekends.