Bus up to mountains near Innsbruck

I can't answer your bus question, but I can tell you about my experience in Innsbruck. I have problems with heights, also, The cable cars used are not the small swingy ones. They are large and can hold quite a few people. It actually is a smooth ride, and I was able to watch the scenery. This probably doesn't help you, but I know what being fearful of heights is like.

Sarah, as a general point, the cable cars start where the roads and railways end. Once it gets too steep for roads, it is cable cars or on foot, and in some cases foot is not even possible. You don't really have a choice of bus or cable car, it is bus (or train) to the cable car base station.
A funicular is a 45° lift (elevator), two cars connected by cable over a wheel at the top, one goes up while the other goes down. It stays on the ground.

Buses (and trains) will get you to many good views, but to really get up a mountain there is no alternative to a cable car - SORRY.

Thanks for the replies! We'll see if going up higher than the bus works out when we're there.

For now I am just wondering if there is a better city bus route you'd recommend for good views? Or is Igls a good place to go?

You can also catch a bus from a square in the city center to Axamer Lizum and catch a cable car up to the lodge on top of the mountain.

The view from the mountain top is absolutely incredible with the same size mountain to the north--and the deep valley below. They've got a restaurant up there, and it's interesting to watch the skiers on what was the ladies' downhill slope in the Olympics..

The whole western Tirol is absolutely gorgeous scenery, and I never tire of the many trips we have made there since 1970.

It's a tram from Innsbruck to Igls - formerly a separate mountain railway but now absorbed into the city tramway.

Bus J to Igls leaves from Museumstrasse, either in front of the Museum or in front of the Sillpark shopping mall. There is also a very central stop near Marie Theresien strasse. The village is nice - I spend my holidays there - but it is a middle mountain destination, no dramatic sights.
Igls can also be reached first by boarding tram #1 to Bergisel and then switching to tram #6 that slowly climbs hills to Igls. This is a really nice trip, again no dramatic sights but very nice ambience. Check on ivb.at if tram #6 is really running as keeping open the line is very expensive and there is a tendence to limit service to high season.

A dramatic sight over the city is from the terrace at Hungerburg. It can be reached by funicular from the congress or, cheaply but slowly, by the same bus J but running in the opposite direction.

Thanks @lachera, I will update my plan to take the J bus to Hungerburg.

I should have mentioned we are traveling in December 2016, and will be in Innsbruck during the middle of the month.