Join a board
For any of those armchair politicos out there, Greensboro needs you to step up and agree to serve on any number of boards and commissions.
While they may not get the kind of publicity that the City Council holds, a lot of legwork is done by boards like the Zoning Commission, Planning Board and the Minimum Housing Standards Commission. While none of the commissions pay their members, it is a way for residents to take a bigger role in how the city is run short of seeking a seat on the council. Many of the current council members also served on a board before they were elected to their seats on council.
Right now, there are 19-odd vacancies on the city's boards that need filling (click here to see the city's vacancy list. The county has their list here). Even if there's not a current vacancy, you can put your name in so it's there when a slot does open up. There are a pool of applicants already for some of them, but the council members who have to appoint people to boards never complain that they've got too many willing volunteers. You can also call your council member to let them know that you're interested. They are the ones that will ultimately decide whether they want you reviewing zoning cases or keeping an eye on the Housing Authority.
If you're interested in a board, you can read more about what they do in the city's Boards and Commissions Handbook. Then send in the application form to the city clerk.