Start by looking up the pedigrees on AllBreedPedigree.com or using the American Quarter Horse Association's pedigree database at AQHA.com. AllBreedPedigree.com provides free simple pedigrees, while the AQHA charges a small fee.
Use the horse's registered name to look up its pedigree. If the horse was a racehorse, it should also have a lip tattoo you can use to look up information about it. If you are just looking out of curiosity, a simple five-generation pedigree may be enough. However, if you are considering breeding the horse, consider paying for a more detailed pedigree that includes performance records and other details about your horse's ancestors.
If you are looking for general information about the breed, you can start with researching the foundation horses. This term refers to early and influential sires and dams that helped create the modern quarter horse. FoundationHorses.com provides free information and pedigree details about some of these famous horses.
There are also books and other educational resources that explain significant bloodlines and the impact of famous horses. You can find many of these through the American Quarter Horse Museum, which maintains a library and archives. The American Quarter Horse Journal also profiles some famous horses and bloodlines, covering a variety of subjects that may interest pedigree researchers.