There was strong interest in starting a genealogy category on WGOM, and I've been sitting on this until I had my surgery to have something to work on while recovering. Since there are a plethora of genealogy related topics, I'm going to make this a stream-of-consciousness introductory post to touch on a lot of things, and entertain ideas for future posts to tackle specifics.
Genealogy is is possibly the second most searched topic on the web (behind pr0n, obviously) and as the years have gone on there has been a flood of data made available online.
There are many aspects of genealogy that intrigue people and get them fired up research their own families, but one of the biggest drives is also one of the most misguided: everyone wants to discover that they're related to George Washington, Charlemagne, or any number of famous figures in history. My recommendation is not to get caught up name-grabbing -- the further back you go with your research, the more chances that mistakes are made. Much more meaningful is to gather the stories and data of your more recent ancestors.
First rule of starting your genealogy research: begin with what you know. Begin with your parents, and their parents, etc, as far back as people can provide. This is your base. Using US census data (now public up to 1940) you will be able to flesh out their families and track their movements. In addition, several states had rudimentary state census "on the 5's" between the US census decades between the later 1800's and the early 1900's. Obviously the further back you go, the more people you will be tracking. I found it most convenient to limit to your direct ancestors and their children, although I would take some side branches down if there were relatives I was close to in those branches.
Second rule of your genealogy research: if you don't already have one, get a library card! There are multiple pay sites which have genealogy data, but most all of them provide free access (some limited, some complete) through your local library. In addition, most library systems have helpful resident genealogy assistants available to answer questions. There is no reason to throw money at Ancestry.com or other pay sites, at least not until your tree has expanded enough to need help.
Third rule of your genealogy research: gather source information. These could be primary sources (birth certificates, death certificates, marriage certificates) or secondary sources (letters, family stories, military service, passengers lists, church records, cemetery markers, etc). In all cases, document where your data comes from! And for crying out loud, get copies of photographs -- it's wonderful to have faces to go with the names. (In the photo above, those are my great grandparents in front, and my mother's father second from the right in the middle row.)
You'll find that your families congregated in certain locations, and it is valuable to get to know the various Historical Societies in those cities / counties. My mother's side spent years in Douglas County MN, and the folks at Alexandria have been very helpful in getting family info, property data, etc. Mrs. Runner had ancestors in Rutland County VT, and helpful researchers there as well as a third cousin of hers had a lot of useful help.
You will want to find a website or software to record what you find. I use an older versions of Family Tree Maker, but if I were to start now, I think I'd create a tree at FamilySearch and have it on the cloud -- the Mormons are leaders in genealogy research. In all cases, be sure to enter your sources when recording each piece of data -- this will help in eliminating questionable entries.
If you are up to it, it's also valuable to have online visibility. I used genealogy as an excuse to learn HTML, and used free RootsWeb (which has been sucked up by Ancestry.com) to host my webpages. I have had countless contacts by relatives (and non-relatives) of me or my wife over the years who have stumbled onto my website via some search engine. It also makes for a nice repository to share info with others in your family who might be doing research as well.
Here are some example sources I have used in doing research on my side of the family and my wife's side of the family.
One thing you should definitely do is set some goals. One goal I had was to take each branch of my family and my wife's family back across the ocean. I've been mostly successful -- my side is pretty much right off the boat, but that includes ancestors from northern Germany 1860-1870 with the difficult-to-search surname Will which has been a tough nut to crack. My wife's side has some fairly longtime US residents in it including Civil War vets from both sides, but she has an Irishman with a fairly common name (James Morgan 1860s) and a Prussian with an uncommon name (Grzmocinsky 1880s). I've been sitting on them, knowing that someday some group's effort to digitize data that will include the smoking gun I've been looking for.
A few useful terms:
GEDCOM -- the standard file extension/format for disseminating a family tree
ahnentafel -- ("name table" in German) this is a numbering scheme for identifying direct-line ancestors and descendants. If you are person #2 in your file, for instance, your father would be #4 (2 * 2) and your mother would be #5 (2 * 2 +1); your father's parents would be #8 and #9.
soundex -- this is a system for organizing names by phonetics to better find matches on names which are difficult to spell consistently. Format is a first letter followed by three numbers; here is a converter to play with. You will find many lists indexed by Soundex for convenient searching
patronymic surname -- think Scandinavian, where the child's surname is based on the father's given name. Torsten's children would be Torstensson or Torstensdatter, for instance. btw, generally you'll find Norwegian and Danish surnames ending with ...SEN, and Swedish with ...SSON. Patronymic naming ended by national decree throughout the 1800s to early 1900s.
Cyndi's List -- Cyndi Ingle started a list of genealogical websites grouped and categorized, and it's pretty much grand central station for finding specific information by location or whatever.
Sorry this has rambled so much, but I wanted to get some groundwork out there to spur discussion. I'd be happy to answer any questions or take suggestions for more specific help.