As the national election comes closer more and more people are discussing politics. People often forget about local government in the discussion. While the federal government sets policy it's often local government that determines how those policies are executed.
First off make no assumptions about municipal bodies. When they meet, who gets to vote in them, and what powers they have are all determined individually normally by a charter. While it would probably be a good idea to read it there are other ways to get a feel for a body. Sit in on a few meetings. You'll get an idea of who votes, which city departments attend and the general style of the council. In addition many cities have "citizen's guides" which explain in general how the city is governed.
Generally municipal bodies are required by law to post a calendar of upcoming meetings and hearings for the public. Most of these meetings are open to the public to observe and even make comment.
Furthermore city bodies as well as city executives often have websites where they list their contact information. My general experience thus far has been as long as you are respectful they will reciprocate.
Levels of Government
One of the difficult things about navigating government is determining what is whose responsibility. State, county, and local governments often overlap in responsibilities and divvy out in the specifics. For instance one road may be owned by the state another by the county and yet another by the city. This is important because it determines who has the authority to do what. For instance let's say that I want to complain that a road has a pothole. I have to know who to complain to. The city government is going to be reluctant at best to maintain a road that belongs to the county and vice versa.
The same also goes for finding out general information. You have to know who to ask. There is no guarantee that the city is going to know the particulars of what the county is doing and that the state is going to know what the city is doing.
Most school boards I know of operate independently of municipal government. This includes collecting their own taxes. They have their own meetings.
Local governments often take taxes in the form of mileages. A millage is based on the property value of your home and various millage rates. A mill is 1/1000 or .1% of the assessed value of a property or So for instance if I had a 50,000 home I would have to pay $50 per mill. Most people have to pay more than one millage.
As stated previously each city is different. The city council may handle the day to day administration of a city or it might fall to an administrator, manager, or various city departments. Also stated previously, in order to solve problems or find out information you need to know who to talk to. My general experience is the best way to do this is to make a phone call. If someone can't answer your question they'll probably point you in the direction of the department that can.
One of the offices that have a good amount of general information is city clerk's office. Generally this department is in charge of keeping and managing city records, including council minutes.
Freedom of Information Act
By law citizens have the right to request city documents. There are various procedures and regulations regarding it but in general, fill out a form listing the documents you want give it to the FOIA coordinator and wait about a work week. You'll at the least get a response and if they deny the request the government is required by law to give you a reason, which must fit within the purview of the Freedom of Information Act's exemptions. To make things quick be specific to which documents you want and give the form to the right person.
It might seem like a lot of red tape and for most things it is, but remember the purpose of the law is to allow the citizenry to have information the government might be reluctant to hand over. Filling out a FOIA request makes it likely that the government will be required to give you the information.