There are quite a few websites out there that can help you get started on your family's genealogy. Before you start out on any of these sites though, the first thing to do is interview your family members! Ask questions, ask for copies of pictures, ask for stories. Always remember to be sensitive...things that to you seem like ancient history that is just an interesting story, could be a very sensitive matter to someone of an older generation.
Ask for names, dates of birth, marriage, death, locations, children, siblings, spouses, parents. If you are so fortunate that your grandparent's are still alive, ask if they can remember back to their grandparents. That gets you back several generations! After you have compiled all the information that you can, check out these sites:
www.familysearch.org - Lots of documents online for free!
www.ancestry.com - Good option for storing your tree online either public or private. Also, for a fee there is access to lots of documents. You can get a lot of good info during the 14 day free trial if you can't pay for the subscription right now.
www.findagrave.com - A growing database of grave information.
www.cyndislist.com - The most comprehensive collection of genealogical websites
It also never hurts to Google some names...you'd be surprised by some of the websites you can find that might have information on some of your ancestors! These are just some places to start...more coming soon!