Completely Free.
I am a staff member of two local Family History Centers and have been for 15 years. It is our job and purpose to help everyone locate their ancestors. No matter who you are or what religion you belong to, and our services are free.

We can assist in locating records to search, and have access to the over 2.000.000 rolls of microfilmed records from all over the world. ( I have personally traced my German ancestry to 1549).

We can teach you how to use the software Personal Ancestral File, and search sources like Rootsweb.com and familysearch.org and to Google for your genealogy.

The local libraries can be found in the white pages of your phone book under Church of Jesus Christ of Latter-Day Saints. Family History Center.

Getting started can be confusing so let us give you a hand.

There are also lessons at Familysearch.org and free home study lessons from Brigham Young University continueing education department

A real interesting thing to do is Google your family history.
Go to google search and type in your name plus genealogy.
Example ("Daniel Boone" +genealogy). Use all varients of your name. Also Google the name of their towns and plus family history.

Have fun.
References :

You can have a look at http://www.new2-geaneology.com
It has lots of links to databases, software etc and many are free.
Also there is a free 'how to' genealogy ebook.
Good luck
References :

The short answer to "How can I find my family tree?" is that if one of your great-aunts has spent 30 years researching it, AND has posted her research on the Internet, you'll find it. If not, you will have to do the research yourself. It is not difficult, but it takes time. Most young people do not want to spend a couple of hours a week doing research, because it is too much like homework. So, you may want to skip the rest of this answer. If not, read on.

If your line has been "done", chance are it is on one of these two sites. When you search, don't fill in all of the fields. Start with given name, surname and birth year. Use (+/-) 5 for the birth year. Expect to spend 15 – 45 minutes on each. Neither has any living people, so don't enter your own name.

http://www.familysearch.com
(Mormon's mega-site. Click on "Search", to start with, or "Advanced Search")

Roots Web
http://www.rootsweb.com
and in particular,
http://worldconnect.genealogy.rootsweb.com/cgi-bin/igm.cgi
(Roots Web World Connect; 460,000,000+ entries, of varying quality)

Here are a few more. The resolved questions have lots of links and tips.

http://www.cyndislist.com/
(240,000+ links, all cross-indexed. If you want Welsh or Pennsylvania Dutch or Oregon or any other region, ethnic group or surname, chances are she has links for it.)

Ancestry.com
http://www.ancestry.com/
(which has free pages and FEE pages – so watch out)
and, in particular,
http://www.ancestry.com/learn/facts/default.aspx?ln=
Surname meanings and origins

http://www.tedpack.org/begingen.html
My own site: "How to Begin"

United States only:

http://www.usgenweb.net/
(Subdivided into state sites, which all have county sites.)
(The Canadians have Canadian Gen Web, by province)

http://ssdi.genealogy.rootsweb.com/cgi-bin/ssdi.cgi
(Social Security Death index – click on "Advanced". You may find your grandparents.)

http://find.person.superpages.com/
(US Phone book, for looking up distant cousins)

United Kingdom Only:

http://www.genuki.org.uk/
(Biggest site for United Kingdom & Ireland)

http://www.freebmd.org.uk/
(Free Birth, Marriage & Death Records)

In the USA, some public libraries have census image subscriptions. Many Family History Centers do too.

This is a general hint: Even though you go in through YA Canada, YA Australia, YA UK or YA USA, all of the questions go into one big "pot" and get read by everyone in the world who speaks English. Most of the people here are in the UK and USA, but you sometimes get questions and answers from people who worry about kangaroos eating their roses. So, if you are asking about a specific individual, put a nation and a state / province. It will help people help you.
References :

http://cyndislist.com/
References :
http://cyndislist.com/