If you don't know how to locate your local health department, then an Internet search engine may be your friend. Searching for either your town's name, or the county and state, and the phrase "STD clinic," will probably bring up the contact information for your local free STD clinic in no time flat.
The web can help you in other ways. HIVtest.org is a government sponsored web page that allows you to search for HIV testing locations by zip code. Although not all places that offer HIV testing are free STD clinics, this site could be a good place to start looking for a local clinic if you can't find one another way.
Not every town runs their own STD clinic. Fortunately, in addition to health department clinics, local public hospitals and teaching hospitals may also run low-cost or free STD clinics that offer both STD testing and treatment. If you live or work near a convenient hospital, you can call and ask if they offer these services, but I do not recommend going through the Emergency Room unless you are actually having an emergency.
There are other free and low-cost STD testing options as well. For those people who live in sexually diverse communities, gay and lesbian community centers and health care organizations often sponsor free STD clinics or offer low cost testing and treatment. There is also a privately run free clinic locator here
Finally, Planned Parenthood is not a free STD clinic, but their clinics often do offer STD testing and treatment at low-cost. Fees are usually scaled according to patient income, and most offices accept health insurance. Remember, Planned Parenthood isn't just for women. They provide treatment to individuals of both sexes.