For “proper” downhill there is the Nordkette trail which you can take the gondola (from Hungerberg) up
http://nordkette-downhill.at/?page_id=5
Otherwise the official trails are marked on maps (Kompass) or I think you can get a brochure from the tourist info, they will mostly be on fireroad though. Officially you are not allowed to cycle on footpaths.
For just messing around, I like the area around Lanserkopf (south of Innsbruck) – it’s quite fun, you can just head up and you should be able to find some good trails to come down on.
If you don’t come too late in the year, a longish route I really enjoyed is up the Stubai valley to Fulpmes then ride the forest road through Schlick and across to Starkenberger hütte, then over the Seejochl to come down the valley towards Adolf pichler hütte and Grinzens. For a long one day or two day trip the Karwandeltour (north of Innsbruck, W-E, ending in Achensee) is good – there’s a couple of variations depending on where you want to sleep and how many metres you want to ride up.
