Innsbruck offers a wide variety of restaurants: Austrian, Italian, Greek, Asian, American, etc. There even are some "exotic" restaurants which have Nepalese or vegetarian cuisine.
You find a huge list of restaurants here. If you prefer a more rural atmosphere, check out the list of restaurants up in the mountains. If you're looking for some really exquisite restaurant for a special occasion, check this list of all Tyrolean restaurants awarded by Gault Millau.
The university restaurant Mensa has several restaurants all over the city centre. You find their daily menu on their website. Access is public, everybody can eat there. Students get a reduction upon showing a special card that you can get at the offices of the student union ÖH.
If you want to know the daily lunch offers (Mittagsmenü) of popular restaurants in Innsbruck, you can either check the respective website of the restaurants or special websites like wegla.net, tagesmenue.at or one provided by the local newspaper TT.
If you enjoy delicious ice cream, you will find several gelaterias in the city centre: Amorino, Eisgrotte, Peintner, Tomaselli, etc. But even if you're more into frozen yoghurt, you can choose between different shops (e.g. fro yo, FYPS, Kurt, Peintner).
Please keep in mind that it is usual to give tips here in Austria. It is not compulsory, but if you don't give anything at all, it might be interpreted as an offence or a sign that the service wasn't good. The amount is variable, but it's roughly 10% or you bring it up to the next round figure (e.g. € 2,90 --> € 3,00 or € 2,40 --> € 2,50).