DNA testing to determine one's ancestry seems to be a popular "thing" these days; you send a DNA sample to one of the testing companies, they process it and give you a report. The cost is not trivial - the $100-$200 range seems to be representative - and the results will take several weeks to come back. the various companies do offer sales from time to time, so you can watch for those.
The actual tests and therefore the potential results can vary depending on the type of DNA test that is done. This article from Business Insider outlines one writer's experience with 23and Me, and goes through the whole process including the kind of results you can expect to see.
I would be remiss to fail to mention that your voluntary DNA submissions may be subject to police requests.