You can purchase an autosomal DNA testing kit from Ancestry for $99 (with frequent sales, so it's worth waiting for one of those). No blood draws -- just follow the instructions and collect a saliva sample. Simple and easy! The AncestryDNA test covers both maternal and paternal sides and is equally effective for men and women. It generally takes about 6 to 8 weeks to obtain results. You'll learn your estimated genetic ethnicity and find matches with unknown cousins. AncestryDNA has a feature called ThruLines® that shows presumptive matches with descendants of specific ancestors through your 5th great grandparents (and may suggest possible ancestor matches). Again, weigh these against research instead of blindly accepting "matches," as the matches are based on online trees which frequently contain errors and unwarranted assumptions.
You can also try FamilyTreeDNA.com, 23 and Me, or some of the other companies offering similar testing. Prices vary, as does client satisfaction, so do your homework before making a commitment to one or the other. I've done FamilyTreeDNA (uses a cheek swab to gather DNA) and got lots of matches plus an ethnicity estimate that was a little different than the AncestryDNA estimate. I found the matches harder to map out on my online tree than the Ancestry matches, but maybe that's just my limitation.
WARNING: Some security officials in the FBI are warning against these kinds of tests because many of these companies partner with entities affiliated with the Chinese government which is building databases of personal DNA info as part of its mission to dominate bio-tech.