The initial start to finding your roots begins with you. Most likely, you know your own birthdate. You probably know your parents birthdates and places of birth. Take that as the leaping pad into your journey.
I started with Ancestry.com. Yes, I pay for it. But they also offer a free two-week trial. For me, the fee is incredibly worth it. Second Snack gives a great review of his experience. I find the site incredibly user friendly, and from what I can tell, Ancestry.com adds more and more resources all the time. Also, when you type in a name of an ancestor, these little leaves start to appear next to the person's name. For me, these little hints cause an almost addictive kind of thrill where you have to keep searching and searching until you find out more and more answers (and more and more questions.) Be prepared to be in front of the computer for a long time; the first time I researched, I stayed in the same chair for five hours without moving.
Family Search is another site I recommend. And it's free! You can begin searching for your ancestors on their search page found here. On thing I really like about Family Search is you can begin your journey by narrowing down your search by country and then by state. (You can find this as you scroll down on their main search page.) I found a lot about my grandmother's family in Tennessee by searching on the state's marriage records.
Many people have misconceptions about starting their family tree because they either don't know where to start, or they feel they already know a lot. Jessie McKinley wrote a great article for Family Share about the misconceptions that can come with researching. Even if you think your great aunt already researched everything you need to know, I guarantee you technological advances have given you even more gems and more avenues of research.
One thing to beware of while you're searching, especially on Ancestry.com: those little leaf hints can link to family trees already established on the site. Not all of the information on these family trees has been verified. It can be incredibly addictive and exciting to take your tree back further and further, but all this information must be verified through records and other established sources. I have a friend who was tracing her tree through another member's tree that claimed (highly unlikely) that Merlin was her ancestor. So just be cautious. Even if Merlin isn't your ancestor, you will find amazing, life changing things.