I used to go to the library and pore through old censuses of places I thought they may have lived which was very tedious. I now have paid for a subscription to Ancestry.com which is awesome!
You must start with what you know about your family, ask older relatives where they lived and where their parents lived what were their names etc, they know bits and pieces of the family history that can many times get you started in the right direction. You can send for birth certificates etc of your great grandparents that many times have their parents names listed on the certificate too. Also look for records (church and marriage etc) in the cities they lived in.
There are also many family surname genealogy sites you can go to (free) like rootsweb and Ancestry.com and try to find someone who may be researching the same people as you are. The surnames sites are all free. Also go to Latter Day Saints site on the internet (its free) and has a lot of info. Warning!! Genealogy is addicting like a sickness you can't stop once you get started.
Flavie