Getting started depends on where you want to go. If you are searching for your Maternal Line, you can request a mtDNA test. I suggest www.smgf.org
. It's free and they will send you a coupon for a paid test from Relative Genetics.
If you want to learn about your paternal surname, get a brother, father, or uncle with that surname or any male relative with that surname and have them take a Y-DNA test.