Genealogy is something that I'm not interested in but my mother, Margaret, is. Over the years, I've built up a little bit of knowledge about genealogical research - both from Internet resources and from my mother.
The Internet provides a quicker way to gather necessary facts to trace your heritage but unless you're extremely lucky (finding a family member who has already done the family tree) you'll still have to do a lot of work. You might want to check out my Genealogy link section.
Getting yourself organised is important too. Using a computer program can be a help but if you don't like computers a decent paper based filing system will do the job too. A computer will help you get organised - not MAKE you get organised!
If you've got an Apple Mac, the best program for genealogy (or one of the best computer programs, period) is Reunion. This has a huge number of easy, intuitive ways to input your family data and number of attractive and practical ways of outputting it. If there's one reason to get a computer (if you're paper bound) it is the ability to instantly create attractive and comprehensive printed family trees - no more graph paper!