The touristy bits of Innsbruck are done relatively quickly. The Hofkirche is a must; it's the mausoleum to Maximillian I (the Habsburgs were resident in Innsbruck for a while), although his enormous tomb lies empty because he was buried in Vienna instead. On past the royal court to the Goldenes Dachl, the famous roof that Max built to impress the plebs. On the square there is the Stadtturm (tower), which is worth climbing up for a view of the town and the towering mountains behind.
Head south to the Swarovski shop, which is definitely worth a look inside, although if you're a fan then you should take the bus to
Kristallwelten about 20 minutes outside of Innsbruck.
You'll find all the touristy shops in the old town around the Goldenes Dachl. The normal shops are just to the south.You can also take the bus out to the Olympic ski jump, from where there's a great view of the town.
Innsbruck is a great place to experience the Viennese Kaffee und Küchen culture,
Cafe Sacher and
Cafe Central being the best. I've never managed to find a good restaurant in Innsbruck. A good pub, however, is Limerick Bill's Irish pub at Maria Theresien-Str. 9.
There's lots of good information that can be printed off at the
Rough Guides website.