Adam Korn simply called the governor's office and then a legislator to see if they were seeking an intern. He suggests trying this method as well as getting involved in state and local campaigns.
The following websites will help students find experience in politics and campaigns.
A list of many campaign-related internships can be found on the Journalism School website at:
http://www.journalism.wisc.edu/career/latest_intern.html
Information about internships for the Democratic Party of Wisconsin:
http://www.wisdems.org
Information on volunteer opportunities with the Republican Party of Wisconsin:
http://www.wisgop.org/Volunteer.aspx
Opportunities with Progressive Dane (an independent political party that believes citizens should control public policy) can be found at:
http://www.prodane.org/get_involved/volunteer/
A full list of state representatives and senators—offices often seeking interns—is available on the state legislaure's homepage:
http://www.legis.state.wi.us/
For information on ways to get involved with UW-Madison's student government, Associated Students of Madison:
http://www.asm.wisc.edu/involved/index.html