It's extremely unlikely that you will find your exact ancestry online -- unless your brother or sister has researched it and posted it on the web!
Start with yourself and then work backwards. Talk to your parents, grandparents and relatives first. Gather all info you can from them.
These sites can help you research your family tree, but they can't do it for you:
familysearch.org
rootsweb.com
cyndistlist.com
usgenweb.com (if your ancestors are in the US)
ellisisland.org
genforum.genealogy.com
There are thousands of location-specific or surname sites as well. Just google "Fayette County Indiana genealogy" and you'll get a hundred hits (just an example, of course.)
Libraries, state archives, state libraries, county courts, genealogical societies, etc. etc. have genealogical info -- more and more of which is online now.
These are all free, but, of course, the subscription sites have more information. Check with your local libraries to see if they have ancestry.com or heritagequest.com databases. You may be able to access these with your library card at home.
WARNING: not every tree you find on the Internet is real or "true." There are some very sloppy, ignorant, or dishonest "users" out there. You have to document what you find with reliable records.
For a good intro on how to do all this, see "How do I get started?" at familysearch.org.