I always start with what I know first. I then use that information in Family search or Heritage Quest to find census, birth, marriage or death records. From those records I can branch out to other people. I've done some African-American research and I find city directories are better sources than censuses or birth or marriage records. I generally do a google search, with the hopes of getting more information. Sometimes, I can find a living cousin and can ask them questions.
I mostly use the free online sources first. I would love to do more research in libraries, but I don't have the resources to travel. So I'm stuck with online sources.