Echoing others...
Old Fashioned
www.theoldfashioned.com - Yeah, it's real Wisconsin designed for tourists. You're a tourist so you'll love it. A better choice though is...
The Great Dane
www.greatdanepub.com - Its Madison's oldest brew pub. There are three locations but the one by the Capital is the original.
Brocach
www.brocach.com - Also on the Capital Square. Irish pub, not a brew pub, but the food is awesome and filling.
Ale Asylum
www.aleasylum.com - Awesome beer. Head brewer has a long history in the Madison area. Not much food, but that's not why you go there.
The Malthouse
malthousetavern.com - This is my buddy's bar. Incredible selection and nice patio if its still warm out. It's on the corner of E. Washington and Milwaukee. The sign still says "Union Tavern" because in the last 2 years he hasn't got around to changing it. He has 18 taps and hundreds of bottles. Beer geeks are always present and the bar staff is the most knowledgeable you will find anywhere. Emphasis is on Belgian beers. This bar and Ale asylem are both on the east side of Madison.
Vintage
www.vintagemadison.com - On the west side, this is our newest brew pub. It used to be JT Whitney's but this is much better.
Capital Brewery
www.capital-brewery.com - Also on the west side, the tour and beer garden are worth the trip just to see. Its technically in Middleton, but who cares. And while you are on the west side...
The National Mustard Museum
www.mustardmuseum.com - You just have too. They used to be in Mt Horeb across the street from the Grumpy Troll brewpub, but they moved about 6 months ago. Huge collection of musturd and you can attend Poupon U. I haven't seen the new digs, but its probably worth the stop.
The Capital is worth a look. Around the square there is the Veteran's museum and Wisconsin Historical museum. Both nice.
From the Capital, you can walk down State street, which is a primarily pedestrian mall that connects the Capital to the U.W.-Madison campus. So on one end you drink with politicians and on the other with students. As you can imagine, the prices drop as you move from the capital to the campus.
The primary section of the U.W. campus
www.vip.wisc.edu has plenty of things to do as well. Not far from the end of State street is the Memorial Union. A great thing to do on a sunny day is to grab a beer and sit out on the terrace overlooking the lake. The building itself is incredible, too. You can also get Babcock hall ice cream. Its made elsewhere on campus, but is about a 1.5 mile walk from the union.
Two pieces of advice, when you are downtown, park your car in a ramp and just leave it. There isn't a lot of parking, so trying to move your car will frustrate you. Too many 1 way streets and road construction.
The other is the "beltline". Everyone will give you directions that start with, "Head to the beltline..." but then you will discover that there are no signs anywhere saying "beltline". The beltline is basically highway 12 and its 4 to 6 lane road that runs along the south side of town. On the east it is hwy 12/18 on the west it is hwy 12/14.
Enjoy. I do every day.