There are a lot of great genealogy websites out there - both free and subscription. But before you go searching online, you'll need to do a little homework first. For privacy reasons, most genealogy sites don't include information on living people (or anyone who 'could' be alive). So you won't find your own name, your parents, and probably not your grandparents on these websites. Most people need to start with their great-grandparents, or someone who was living in 1930, before genealogy websites can be helpful.
So start by talking to your family members - especially the older ones - and see what they can tell you about your family. Don't forget about your more distant relatives like great aunts & uncles and cousins... they may have bits of the story that your immediate family doesn't. For each ancestor, try to gather the following info:
Full name (maiden name for women)
Date and place of birth
Date and place of marriage
Date and place of death
And of course any details about their lives, jobs, personalities, adventures, etc.
Once you've gathered all the info your family can give you, you can begin looking for historical records to fill in the blanks and take you back further. No matter how average or nondescript your ancestors may seem, they did things throughout their lives that got recorded: they got married; had children; held jobs; bought and sold property; paid taxes; talked to the census taker; perhaps served in the military, or immigrated from a foreign country. They passed away, leaving wills, death certificates, obituaries and cemetery records. All these records contain bits of information that can tell you about your ancestors' lives, where they came from, and who their parents were.
Not all of these records are online, but many are. Ancestry.com probably has the largest collection of online records, and there are lots of other great sites as well. Here are a few of my favorites:
FamilySearch (the Mormons)
Aside from historical records, Ancestry and other sites allow users to upload their family trees. If you're lucky, someone else has already researched part of your family and published it to one of the online databases. But keep in mind that these online trees were not created by Ancestry employees or by a magic computer - they were submitted by regular users like you and me. Some may be experienced researchers, some may be beginners who've made some errors. If you find your family in an online tree, be sure to verify the information yourself by searching for historical documents to back it up.
Another way the internet can help is by getting you in touch with people who can help you: cousins who have researched your family, experts in specific areas, people who will volunteer to look up information for you. GenForum, Ancestry and Rootsweb have message boards and/or mail lists for specific surnames, locations and research topics.
The LDS (Mormon) Family History Center in Salt Lake City has the largest collection of genealogical records in the world. It's a great place to visit, but you don't have to travel there to utilize their resources. Most Mormon churches throughout the world have a branch of the library inside their building: a Family History Center. You can borrow records from the big library on microfilm and have them sent to your nearest Family History Center. The volunteers there are very helpful. You don't have to be a member of the church, and they won't try to convert you.
Hope this helps some! Good luck!