Innsbruck. I've been there twice in the winter. Actually, I was in Igls, up the hill from Innsbruck. Very cool area! The Patscherkofel ski area is right there. In fact, one of the ski runs actually goes OVER the finish area of the bobsled track. On your daytrip to St. Moritz, definitely check out the bobsled track. It is the only one that is manmade from scratch using blocks of ice each year. When I was there, they had little Viewmaster-like things at different places along the track. You looked into the lenses and saw that that area looked like in the summer. Really cool. It's also uber-expensive in that town!!! And, it's hard to find a relatively cheap place to eat. There is one pizza place over by the track that all the athletes would hit up.
Another daytrip you could do from Innsbruck is to Garmish-Patenkirchen. Another town with an Olympic history.
For packing, I made do twice for 2 week trips and one 3 week trip using one rolling duffel bag. Now, I wasn't skiing, so I didn't have to worry about that stuff. But, everything else would be what you would take on a ski trip (I was at the tracks all day, so I had to dress just like I was going skiing, just with big outdoor boots instead of ski boots and skis). Work it out so that you can wash out a turtleneck every night and have two of them already dry. Keep it to neutrals (white, black, grey) so they will go with everything. Bring your favorite gloves/mittens. I would actually wait until I was over there and go shopping for some of the little things, your hats, scarves, things like that. They are commodity items and can end up as good souvenirs of your trip. There are a couple of decent sporting good shops in downtown Innsbruck. And, definitely pick up some things in St. Moritz. The little things will cost the same as anyplace else.