I have been doing genealogy research on my family for over 20 years and I started with my parents and my grandparents.
First, gather all the information you can about your direct ancestors...parents, grandparents, great-grandparents, etc. Collect birth/death/marriage dates and locations. Record the source of the information so you can refer back to the source if there is a discrepancy or you need additional information.
Ask family members for information, photos, obituaries, personal Bibles, journals and diaries, and anything else that might contain personal information about your family.
Get genealogy software so you can keep all your information organized and in one central location. I have been using Family Tree Maker for over 15 years and it works great for my needs.
After you've gone as far back as you can using word of mouth and family recollections, start using internet resources. There are genealogy sites all over the internet, as well as census records, birth/death records, and other personal data sources. Visit your local library and consult their genealogy section, if they have one.
That will get you off to a very good start. :)