I've been researching my family for about 8 years now. Mainly on my mother's side as my cousin, who is a professional genealogist, is doing my father's side....although I've also done work there as well. Many of my cousins are also tracing out different lines within the family tree. Allows for some very interesting conversations!!!!
To start, get yourself a good genealogy program. I'd recommend a couple. PAF (Personal Ancestor Finder) is a free one from the LDS, Family Tree Maker (which I use), Legacy (which I also use), Hereditus (for Macs), MacFamilyTree (also for Macs), Roots Magic and there are a number of others.
Join a Family History Society...and not just in Australia. Especially if you have ancestors from the UK, try and join the society in the area where they came from. They'll help you heaps with looking for the information you need. They're usually not all that dear to join and I would highly recommend that you do so, if you can afford it.
Now, for the net. There are some very good sites, and there are some pretty atrocious ones as well. Some are free, but many expect some payment for their services. FamilySearch.org and Ancestry.com would be good places to start, however be careful, and this goes to all sites on the net. Not all the information they provide is accurate, or even factual. There is a lot of rubbish out there....quite a few false genealogies have been floating about for a very long time. Many people seems to think that collecting names from other people and adding them to their trees, just because they all seem to point to the same ancestors, is genealogy. It isn't, it's just collecting names. It's exceptionally prevelant amongst people from the US, looking into their family's history. Elsewhere as well. I've seen trees with as many as a million names or more. No one could research that many names and connections within a lifetime of work, let alone a few years or so. You'll also see many, many genealogies (manly US based) purporting to go back to Adam and Eve, Abraham etc etc. It's all nonsense and just pseudo-academic tripe when they say it's all thoroughly researched and definitive. I'd avoid them like the plague.
Now, it's entirely possible to be able to trace your ancestry back to Edward III, Charlemagne, Alfred the Great, Lady Godiva, Robert the Bruce, William the Conqueror or any number of famous historical persons. Quite feasible and has been done...you don't have to be royalty or from the nobility. It helps though....what you need to find is what they call a "gateway" ancestor. Someone in your past who is either of note or part of a family with noble/royal connections of some kind. But with anything you do in research, try to gather as much information from primary sources as you can. That doesn't mean FamilySearch or places like that....especially once you generally get beyond about 1700. What you need to do is to familiarise yourself with census records, parochial (parish) records, probate records (wills, marriage patents/certifcates etc), court records, the various Rolls issued under royal warrant (like the Close Rolls, Patent Rolls etc), BMD records (in England and Wales from 1837 onwards, Scotland from 1855 onwards). Books on peerage like Burkes, The Complete Peerage, Debretts etc, can be of use to you if you have notable ancestors, but keep in mind they can be not so accurate themselves at times. Many genealogist and historians have written various books over the years which can also be useful, but like with everything else you need to be discriminating with the information. It's generally pretty good, but sometimes you'll see where things don't add up with what they write. You will find that you can't always rely on obtaining primary sources of information as in many cases they don't exist anymore. So you have to become part detective, part scientist, part "soothsayer"!!! at times.
Most of all, just have fun compiling your family history. And the one thing to be absolutely diligent in doing....ask the family about anything they know!!!!. Get everything down before it's too late, try to keep good notes and back things up religiously. Don't keep it all on computer and/or disk.
If you need any help or other advice, websites to visit, even some goodies

, give me a PM and I'll be more than happy to help