Build a good foundation for conducting genealogical research. Follow a systematic process:
1. Interview your living relatives and family friends. But don't take what they say about others as gospel truth. Stories get mixed up.
2. Examine your documents and those of relatives that will allow it.
3. Prepare for research by learning about basic genealogy, genealogy specific to your known ancestors. (What documents would exist for where and when they lived?)
4. Organize your data. Free software is available.
5. Research one document at time for one generation at a time, one ancestor at a time. Record only facts found in the documents.
A few hints for making good use of resources: Because of privacy laws and etiquette standards, you'll want to get back 70-100 years before hopping on the internet or trying to order records. AVOID user-submitted or prepared trees/pedigrees except where they have sources cited, and go to the sources. Seeing the same information multiple times in trees does not mean it's more accurate, just that it got copies a lot. Indexes and transcriptions are better than trees but still likely to have more errors. Use these to get you to the original source. Try to find and work with original documents or images as exclusively as possible. Understanding how each type of record was collected and prepared and what that means is an invaluable tool for assessing the information. Not everything is online and/or free, but you can get a great running start for free. Once you get going you'll find there's plenty to work on.
For step 3 I like to recomend getting involved in family history fairs, or attending lectures. You might find them through your local Family History Center, public library or community center.
Some helpful starting places:
http://www.cyndislist.com/ (START with How To and Genealogical Standards and Guidelines, they also have guides and links for specific places)
https://www.familysearch.org/
http://www.censusfinder.com/
http://www.findagrave.com/
http://www.deathindexes.com/
Here -- Many folks that hang out over in the Genealogy section have resources or knowledge specific to certain documents, times, places, and groups.
Google -- Look for local libraries, archives, agencies, and GenWebs, or websites similar to above for your ancestor's country.
Offline -- Libraries, archives, museums, church records, genealogical societies, Family History Centers (operated by the Mormons), etc. BIG note: FHCs and some libraries have FREE access to some of the paid subscription sites like Ancestry.com and librarians or volunteers that are happy to help.