Good Places to Start Your Research

You can try simply putting your oldest known relatives into a search engine, you might get lucky and find them in someone's genealogy website.
The best free search sites are familysearch.org this is the Mormon church's records and rootsweb.com they have the same family trees as ancestry.com. You can also search for free the 1880 and 1900 census records at ancestry.com for other census records there you'll have to subscribe.
If you know the state and county your ancestors lived in that is an excellent place to search either online or with a personal visit.

Suggestions on how to start

First interview all living relatives. Remember that not all the information you get from these interviews will be correct, people's memories are not perfect and some like to improve on an interesting story, so use public records to verify. These also are not always correct the information still comes from people and is sometimes misunderstood, written poorly, misspelled and put in the wrong place. But if 2 sources agree it's most likely correct. Once you have the information on your parents, grandparents and as far back as living relatives can take you then you are ready to start researching.

Just a thought

Remember the people that came before you, and think of the ones that will come after you and how they can benefit from this work that you do.