I have been doing it for years - it used to be lots of hours reading through microfiche, writing letter and then Family researchers invented the internet to speed things up. Net made it easier

- there are lots of user groups (yahoo types) around the place that have been in existence since yahoo started. They normally cover regions.
You should rarely have to pay very much anywhere as most researchers "love" to share their info. Normally paying to use databases is because someone went out and copied/transcribed public database and so there is cost recovery and profit - but libraries normally have it all or Public Record offices, or family history groups
The best online database is
http://www.familysearch.org/ THE LDS are big into family history and have spent years collecting datas around the world. Since they opened this database I have made so many connections
One thing that may be hard when you first start on a psite like LDS is YOU or your parents may not be in it - as there are normally rules and software settings not to publish living people or the last 90 years (for example)
A great Links page is
http://www.cyndislist.com/How to start - look at something like
http://www.cyndislist.com/howtotut.htm#RootsWebApart from that: start with parents births death Marriage certificates - then grandparents - and work back. You will normally find it easy to start and you will be back a couple of hundred years in no time. But at some stage start reading and trying to put context to it all. If you find a region where your clan came from - try reading books about the area - other then give you some new knowledge it can help fill in blanks.
Also don't believe what you find on the net or in libraries as being true. It might sound real and good but Confirm every step /fact yourself. Sometimes the data was collected ages ago, and has errors or mistakes
So what are you still doing sitting here - start researching - but one more link before you go is
here