I would highly recommend visiting State street and the downtown area. There are tons of neat little shops and galleries on State Street, as well as several smaller museums and just about any kind of food you could possibly want to eat. Not to mention that many of the people you'll see there are a sight in themselves. The capital building is amazing, and is open to visitors during the day. There's an observation deck around the rotunda that may be open that gives a great view of the lakes and city.
It doesn't sound like you'll be here on the weekend, but if you are, I'd suggest checking out the saturday morning Farmer's market around the capital square. Its one of the largest outdoor markets in the country. Its a pretty neat experience even if you don't intend to buy much of anything.
If you like plants and flowers, you'd enjoy Olbrich Gardens on the east side of town. The outdoor gardens are free, and the tulips, daffodils and other spring flowers should be blooming at that time. The tropical conservatory is absolutely beautiful and only costs $1. Its one of my favorite places to visit (and to take pictures!)
I'm not too familiar with the university area. I've heard the main union is a neat place to be. There are some nice walking trails around the lake or within the UW Arboretum if the weather is nice. The Hilldale mall is right on University Ave, and West Towne is not too far away.