This is inspired by a former education professor of mine who is big on teaching government beyond the federal level, since the local and state levels are where we have most of our personal encounters with government.
A good starting point is Lessons On Local Government. (With thanks to the instructor mentioned above for the link.) Note the sponsors of the site, and consider the role that Special Districts play in our day-to-day lives.
Two sites that will help you keep current on the relevant issues facing local and state governments are RouteFifty and CityLab, both parts of the The Atlantic family of publications. CityLab deals specifically with urban issues; RouteFifty has a broader focus. Both have daily newsletters that’ll give you a quick look at what they’re covering.
Check out ‘None Of These Votes Are Easy’: Learning The Ropes On City Council, which is from NPR’s Been There (“lessons from a shared experience”) series. It was one of the best government stories I’ve heard in some time. (I understand that the above story is also on the Lesson On Local Government Facebook page.)