Go sledding at Nord Park. You can rent these little plastic sleds for 3 euros an hour. They were kids ones but they do the trick for any big kids who are looking to have some fun. Climb up some of the higher hills without getting on the chairlift and slide right down the trails. You just need to watch out for any snowboarders or skiers that are coming down. If you go in the later part of the day - around 4pm there is less  people around.

In 2007 they replaced the old Hungerburgbahn (from the city up the mountain to the Hungerburg) with a new modern railway that has four stations: the Congress - Löwenhaus - Alpenzoo - Hungerburg. Make it a day trip. Take the Hungerburgbahn up to the Alpenzoo, where you will be amazed by the sight of so many different animals from the Alps. It's the highest located zoo in Europe and it's definitely a very special location. You'll get to know the wildlife of the Alps, from otters to wolfs to the famous "Murmeltier" (marmot). It's open all year round. In summer it's really nice to sit on the terrace of the small restaurant and have a drink or an icecream there while enjoying the sight. When you're finished, take the Hungerburgbahn up to the Hungerburg, a lovely village which grows every year. From there one, it's up to you what to do. There are lots of options. Like suggested above, in winter it's really nice to go to the Nord Park (take the Nordkettenbahn - the gondola lift - up to the Seegrube) and do some skiing there. You could also go in summer and have dinner at the restaurant or just enjoy the beautiful view of Innsbruck you have from high up the mountain. You can also choose to go for a walk in the surroundings instead, there are lots of wonderful walking paths that go up to the Alm restaurants (Almhütte) on the mountain, from short walks to very long ones. There are signs which will show you the way. If you don't want to go up the mountain, you can also take the path that leads you to the Gramart restaurant (about half an hour by foot) where you can have some delicious meals for not too much money. You will definitely not regret visiting the Hungerburg. You'll be amazed by the beautiful view you have from up there. Definitely a must see!