I totally disagree with RoterStern here and love Innsbruck. I’d go as far as to say it’s one of the places I would happily leave the UK to live. I’ve spent time in a few other Austrian cities too and this is one of my favourites. Certainly way less sterile than I found Vienna. My wife’s cousins lived there for a while too so got to see a few of the “local” rather than just tourist spots. Maybe that’s helped shape my opinion.
The old town in summer is lovely. You can meander around around with the mountain back drop – loads of pedestrian areas with outdoor bars and cafes. Last week they had an international long jump competition (a bit random) in the city centre.
Take the funicular from the city centre – 1st stop (Hungerberg) you have the Alpine Zoo which will amuse the children (the older one at least!), and you can even feed the goats by hand.
Up one more stop from there you can pack the downhill bike and come down the Nordkette
Shopping Streets and malls are very good.
TBH I have not biked in Innsbruck but cannot see why it wouldn’t be good. There are certainly loads of people in the city centre – usually hanging out by the river having just come off the hills on various trail/enduro set ups. But you can also jump on the train and do any of the nearby valleys (Oetztal (Soelden), Pitztal, Ischgl, St Anton, or even head on to Lichtenstein for a day (2 hours away). I’ve only biked in St Anton (1 hour away) – it’s more XC but they are ramping it up each summer. There is a little bike park there too and lots of rentals on offer. Lots of the mountain huts are open too with the lure of dumplings and beer as a reward for your climbing.