Those of us who have been involved in genealogy research for a while know how rewarding (and addicting) it can be. To do it right involves a lot of time and effort, but you don't need to be an expert to get started.
...Family, of course! If your parents or grandparents are alive, go talk to them as soon as you have free time. First get the basics, if you don't already know them. Take a pad of paper and ask them their birthdays, marriage dates, maiden names. They will usually know important dates for most people in the family. Write down birth places, religions, Churches family members belonged to and how many kids they had. At this stage you just want the basic information.
It's important to realize that no matter how much you love your family, they probably got mixed up somewhere. Take a trip out to local cemeteries where family are buried. Ask the people at the funeral office if they can print you off obituaries or information. It's a little known fact that cemeteries are digitizing the database of information they have collected on the people buried there.
So now you have some information written all over various pages, right? It's time to get organized! All your family tree research is useless unless there is some order to it. If you don't have a lot of names, I find it's easiest to grab a fresh sheet of paper, write the surname of a family tree branch at the top, and draw up a tree. Start with yourself and go back, focusing primarily on the parents and your direct relatives for now.
Doing so will help you visualize the family tree stretching back a few generations. If you have some money that you're willing to spend, invest in a computer program like Family Tree Maker. This is much better and easier to use than paper, and you can include all sorts of information as your family history begins to take shape.
Online Genealogy Research
You're organized, you have basic information, and now you're ready to get into the really detailed stuff. Since you're just starting you are probably not willing to dish out an obscene amount of money for a membership to Ancestry.com or anything like that. So start off searching the message boards located on Ancestry.com, Genealogy.com, Rootsweb.com, and the like. Chances are, someone in your distant family has already done a lot of research. Understand that information may be incorrect, but over time your skills will develop and you can weed out the false information. There are personal sites devoted to surnames that are useful, and Familysearch.org is a great tool that many use to get started.
When you are ready to take the next step, try a free trial from a major genealogy website and dive into Census Records, Military Records, Court documents, Ship Passenger logs and anything else that may pop up.
If you have money and time, you can begin to actually travel the world, following the steps of your ancestors and find documents and places firsthand. This is rewarding, but not many people can go to this step. I know I've never been able to travel very far in this pursuit.
Good luck and have fun with your new hobby!