Learning of your family history is both a fun and rewarding pastime. Genealogy of course isn't anything just invented as a hobby with the arrival of the home computer but it sure is a lot easier. My Mother was doing genealogy research in the early 70's. She was a stay at home mom. The kids were older and didn't require as much attention all of the time. She would go to the local library on the weekends poring through old newspapers and ordering through the local library microfiche (archived film of records) and other materials to go over during the week while the kids were in school. These methods still work but have been greatly enhanced with the computer.
First off you need to concern yourself with storage. I recommend you get a commercial piece of software such as Perfect Family Tree or some similar program. You can get a free program but I recommend a paid program because you have somewhere to tun to for help in using it. Also all of your valuable information is stored in a defacto standard which will make things easier for you as you progress in your hobby. You should go ahead and study how to use this program before you ever get into your research.. This will save you a lot of time in the future. Don't worry, just an hour or so is all that will be needed. As you use this program it will become second nature to you.
The first step to your research should be Cyndi's List (http://www.cyndislist.com/). Just as the name suggest this is a list. It is however more then just a list. It is hundreds of list that covers virtually all area of genealogy ranging from beginners all the way to expert. These are links that will send you to the needed area of interest. You could spend years just reading what is here which would be a big waste of time. Just start with the links for beginners. You will find out what you will need both in time and knowledge. You may decide just from reading what is involved that you have neither time or interest for this hobby. Most likely you will become entranced with your new interest. Some of the links use will be directed to will be paid and some will be free. The vast majority of these are free. I never use the pay for any information. All of the following sites are free and are the ones I use.Geni (http://www.geni.com/) Is a good place to start. Just start adding the name of the family that you know. This is a good place to start. If you don't know where you are going you don't know in which direction to start walking. Head of to Familyeducation (http://genealogy.familyeducation.com/family-names-surnames/meaning-origin ) This site isn't a genealogy site but it is fun. Find out the history and meaning of your name. Think of it as a source of interest for your hobby. Now head over to GeneaSearch (http://www.geneasearch.com/tools.htm ). This is an excellent source of information. Think of all of these sites as power tools in your toolbox.
Remember what I wrote about Mom doing research at the local library. Well that was using the hand tools of research. Now you can visit any library from the comfort of your home. Get the library card for all of the libraries in your area. Then you will have access to all of their resources on genealogy on line. You will find The librarians are a great source of where to find information. Let us not forget newspapers as a great sours of information. Time machine (http://timesmachine.nytimes.com/browser ) is where you go to get information in old New York Times. Do you have black relatives then give The USF Africana Heritage Project is a good place to start.
Well now you have the basics so start your research. I think you will find that old story about how that old trunk came on the Mayflower may turn out to be just a family legend and nothing more so be prepared for the truth. Keep all of these sites as well as any others you wish to go back to in your hobby folder under genealogy. You don't want to keep looking up the same web site or regret forgetting where you found that picture of an ancestor in a local paper.