Inner city
Rathaus Gallerien
(http://www.rathausgalerien.at/), 2 minutes walk from the old town, main entrance via maria-theresien-straãe
Kaufhaus Tyrol
(http://www.kaufhaus-tyrol.at/), recently opened shopping mall with five levels, located right in the inner city just opposite rathaus gallerien.
Sillpark
(http://www.sillpark.at) just outside downtown--turn right from train station (hauptbahnhof, walk one block, turn right, walk under a railway--and you are looking at it. all major bus and tram lines take you there.
Outside down town=
DEZ
(http://www.dez.at) bus lines c, s and t, many stores just right around it, such as ikea etc.
Cyta
in the suburb "völs" bus line t
Furthermore, there are several warehouses, especially in the nearby village Neu-Rum.
Shopping areas: There are numerous shops in central pedestrian areas like Maria-Theresien-Straße, the Old Town, Franziskanerplatz, Sparkassenplatz and Anichstraße as well as Museumstraße. You will also find shops/stores in quarter centers of Wilten tram lines 1, 6 and STB and Pradl tram line 3.
Souvenir stores in the Old Town offer souvenirs of varying origin, but the Tiroler Heimatwerk Meranerstraße 2 offers real Tyrolean handcraft. However most of the shops are real tourist traps and are overpriced by far. You will probably find more authentic and cheaper souvenirs in one of the surrounding villages of Innsbruck.
Malls: There are several shopping malls in Innsbruck: