Searching Online Records
Family history record sites generally provide the same basic search fields, with some slight variations or additional search and filtering properties. Start by choosing one ancestor from your pedigree that you have some personal information for, such as their birth year or a location. Unless your ancestor has an extremely unusual name, you will need a little more information to narrow your results.
If you don't know the birth year, go ahead and estimate it. Most sites offer you the ability to search within a range of specific dates, or indicate an offset of anywhere from 1 to 20 years. This will narrow your results to individuals with the same name born within the indicated time period.
Know the state, but not the county? If you have an idea of what part of the state the person was born or lived in during their life, you can open a county map for that state in a separate browser tab so that you can compare it to the counties in your search results.
For example, if you know they lived in East Texas, open up a Texas Counties map, and look for counties in the eastern part of the state. You can find state county maps at the usgenweb.org site for the given state, or just search google for [Name of State] county map (or Parish map if the state is Louisiana).
Unless you are looking for a specific record, keep your searches fairly general. If you enter too much data, such as spouse and parents, you may end up filtering out records that don't contain this information.
Experiment with different search combinations, such as the birth year, location, and father's name to get results for different types of documents. Or search just the last name with a residence location and date range to return all records of that surname living within the given area at a particular point in time. You can also just search the parents' names to find documents relating to their children.
Many people go by nicknames, their middle name or even just their initials. This was no different in the past. One of my ancestors bears a different first name or initials on just about every census he appeared on. Be sure to search common nicknames ("Bill" or "Will for William", "Mattie" instead of Martha, "B. F." for Benjamin Franklin).
Also, when searching for female ancestors, run separate searches for maiden name and married name. If you don't know the maiden name, it can usually be found on birth and death records for the ancestor's children, her own death record, and on marriage records.
No Image Available
Some results will be accompanied by a scanned image of the record, but not all of them. In this case you will need to record the available data either in a .txt file (or whatever word processing file type you are familiar with), or in a notebook. This information will provide clues for further research, as well as information to help you find the physical record at a genealogy library or other history archive.
Online Family History Record Sites
This is hardly a comprehensive list, but it is enough to get you started.
FamilySearch.org - A massive collection of digitized family history documents. Completely free to use.
Ancestry.com - Possibly the largest collection of genealogical documents. Some collections are free to view, others require a subscription.
HeritageQuestOnline.com - Can be accessed online if your library is subscribed. Go to your library's website and search their electronic resources for HeritageQuest to see if it is available to you. Have your library card ready.
WorldVitalRecords.com - Requires a subscription.
Feel free to add your own tips for finding family history documents online in the comments!