How to Start
Links to 11 great sites
Grandfathers in Queries
Beginning Internet Skills
Off the web
People after 1900
The LDS 1880
Googling your Ancestors
Main Genealogy Page
Essays on Genealogy
If you are looking for someone born after 1900, you have advantages and disadvantages. One advantage is that most states required birth certificates, death certificates and marriage licenses by then. One disadvantage is that ethical genealogists omit details on living and recently living individuals, to preserve family privacy.
To start with, if the surname is rare, try a phone book. Two of my favorites are Any Who and Verizon's Super Pages Both offer searches by name across the whole country, within a state, or within a city. You might find the person, if he is living and doesn't have an unlisted phone number. Even if you don't find the person, you may find a relative.
If you are sure, or suspect, that the person has passed on, try a Social Security Death Index. The Mormons have a free one: LDS SSDI. If you find the person, you can write for a copy of their Social Security Application. It costs $27 and takes 6 to 8 weeks. That will get you the person's mother's maiden name, which always helps. It will also have the father's name, and the person's birth date, birth place, address and employer at the time.
If you know the town the person died in, you can write to the public library there to see if they have a copy of the obituary. Obituaries will often tell you where and when the person was born, names of the parents, spouse, siblings and children. The smaller the town, the better the chance the local paper wrote an obituary. The librarian will need the name, exact death date and a self-addressed stamped envelope (SASE). I always send a $5 donation, asking them to use it for copying costs and to donate the balance to the Children's Book Fund. About half the time I get a copy of the article. If they don't make copies for out-of-town people, they return the check. You can use the yellow pages section of the telephone book sites above to find address of the library. My page Obituaries and Etiquette has more on that subject.
If you know the state where the person was born, got married or died, you can write for a birth certificate, wedding license or death certificate. Each state has slightly different rules and costs. Some states keep the data at the state level, some at the county level. You can look at the state level page on US Gen Web Most of them have a link labeled "Vital Records". You can also go to Yahoo!, click on "US States", then "Government". Find the official site for the state and poke around. The vital records are often in the Health Department.
Funeral homes and cemeteries sometimes have genealogical records. The obituary may tell you the name of the funeral home and/or cemetery. You can look up cemeteries and funeral homes in the yellow pages in the telephone book sites above. They will need the person's name, exact death date, and a SASE. I send them $5 to cover copying costs too, asking them to donate the unused balance to the staff donut fund or the GAR decoration fund. (The Confederate Veterans Decoration Fund, if they are in the south.)
If the death was recent, the home or cemetery may have the name and address of the person who made the arrangements. They will not send you the name and address, but they may (MAY) forward a short note to the person. This time you send them a blank, unsealed, stamped envelope, the note and another $5 for the donut fund. They read the note to make sure it isn't a death threat, then put it into the envelope and send it.