Visiting a cemetery is a very in-person and hands-on way of collecting family history information. You can find graveyards with your ancestors and other family members through various online cemetery records, print or online obituary announcements, or from still-living relatives.
Tombstones and grave-markers can give you a lot of important information about the people who are buried there. Birth and death dates are the most obvious, and are found on nearly every stone you’ll find. Symbols on the stones may indicate church associations, which can help you locate further records about them (often birth, death or marriage records).
Spouses are almost always documented on gravestones, possibly with birth/death dates of their own. Children and siblings are also commonly found, at least by name. If the deceased was in the military, you can find their rank and unit information on some stones, which can be very important when searching through any other military records. Membership in other social or fraternal organizations (such as the Knights of Columbus, or the Freemasons) may also be recorded on a gravestone.
Also, check nearby graves in the cemetery for other closely-related family members you haven’t yet discovered. But always take note that they are only possible relations, until you research further. Having the same last name is a good clue, but not a definitely piece of proof alone.
Tombstone rubbings have long been the traditional method of recording what you’ve found. A large piece of white paper is held up against the stone, and rubbed with charcoal or chalk to create an image of the carving. While traditional, it is a time-consuming task and large rubbings can be hard to store once you get home. An easier method is just taking a photograph, especially when dealing with very old (and unstable) tombstones. Inexpensive digital cameras are perfect for this. Don’t forget to check the back of the stone when making your record.
More than anything else, always be respectful when visiting any cemetery. Walk around, not over top, the graves themselves and don’t disturb any visitors who are also there. Though if you see someone visiting a grave you are interested in, you can introduce yourself and you may find some new relatives that way.