This is always the most crucial step. Where do I begin researching my family? For most people, they know their parents names, and their grandparents names. If you do, this is where you want to start. I recommend purchasing some family tree software to keep track of everything. Personally I prefer Family Tree Maker.
From here, if you're lucky, you have parents/grandparents/aunts/uncles who are still alive and you can ask them about their parents/grandparents.
If however, you don't have anyone who knows who your grandparents parents were, the next step is to start digging into records. There are many different avenues you can start on, all with their own pros and cons. If your grandparents were born pre-1940 and you know where they lived, census records are a good place to start. The only problem you may come across is if two or more people of similar ages with the same name lived in the same area, or if you don't know where your grandma Mary was born or what her maiden name was.
The next place to check are your grandparents obituaries. Usually they will include next of kin, which often times will produce their maiden name (if they are female) and any siblings that were alive at the time of their death. Sometimes they will even include their parents names right in the obituary. Generally speaking, the more recent the obituary, the more information provided. Generally, obituaries and old newspapers for your area can be found at your local library. If however they don't have a history center, check with your local historical society as they will likely either have the newspapers on microfilm or know where you can find them.
If the obituaries you find are less than helpful, death records can often be obtained from the state. For example, in Pennsylvania, death records can be ordered for those who died post-1906. And recently, they made their index for 1906-1961 available online. This is exceedingly helpful in cases where one doesn't know the exact date of death, as most states will charge extra if you need them to search over a time period. If you're interested, you can find the indexes here:
Often times, parents names and locations of birth will be included on these death certificates. The only thing you might have to worry about is whether the person giving the information is correct in their assertions.
Normally these sources are able to get you back at least one additional generation, if your grandparents were alive in the 1900s. If not, other sources must be checked.