The less information you have, the more work it will take.
There are wonderful websites that can help you, which are free for the first 30 days or so. You should buy a computer program for this, or you can just start with a large family tree on paper. If there is a Mormon church near you, they have a family history center which can be very helpful. It is free, and you don’t have to be Mormon.
The best first step, though, is to talk to family and get all the names and dates and locations you can. Once they are gone, it can be quite difficult to do this.
Good luck!

You have to pay at ancestry.com but they have a lot if info. Census records, burial records, military records. I paid for a month. There was do much info it was hard to wade through.

You do not have to PAY! If you go onto the Forums of these websites!
(Rootsweb.com allows this after you register a user name etc to allow contact from those that
are interested in your lineage).
IF you have your Grandpatrents names and dates then go order a copy of there marriage
certificate to obtain original birth place and Great Grandparents names too.
IF you do not have that info to start then obtain your parents Marriage certificate,Birth records etc
Much can be accomplished this way and it is highly recommended as you may need to list the sources that you received the info from (verified true pedigree startup).
Do not place this info on a software Family Tree Program UNTIL ALL SOURCES ARE
VERIFIED as true to you line.
I found that in two different programs I could NOT EDIT errors AFTER I saved?
Lots of info for free on www.collectionscanada.gc.ca if researching Canada records. It gives aquite a few tips on how to go about searching online too.
Just Google it in your search engine for Genealogy Archives and name of country.
Another beginners guide online is www.about.com
It is the journey not the outcome that makes researching fun.
Have fun and IF and When you gather all info plus photos from your parents albums and Grandparents albums go onto www.blurb.com and MAKE your own Family Book .

You can buy programs that will allow you to construct your own tree without having to pay an online fee. I would go that route at first, begin to get yourself an outline of your family and, if at some later date, you decide to go the Ancestry route, you can always download the gedcom you have constructed directly onto the site.
That being said, sites like Ancestry have tied up a whole lot of information that used to be free and made it for purchase so at some point you will end up doing just that.
Another good source that you can use, and it is still free at this point, is the Mormon Church records, they have quite a bit of information that will probably get you started. You can find them at https://familysearch.org/search. This is actually a great entry portal for your search.

You could do it the old fashioned way to start with. Find an old relative with a good memory that can tell you everything they can. Sometimes they keep contact info on more distant relatives. Contact them and then get everything they know. This will give you a lot more to go on with the websites.

Good suggestion @keobooks. I have found I gather much more information from older relatives just by getting them to talk about their lives and guide them with leading questions. If you ask a bunch of questions at one time they just get flustered and can’t seem to remember as much. I think it kind of puts them on the spot.

And @keobooks way lets you get the stories that you won’t find in databases.

geni.com

You can sign up for a free ancestry.com account and cheat like crazy on all their “tips”. In Canada they tell you where they found the info (usually official government sources). Just go directly to that site and search for your ancestors. Old war records are great as they list next of kin which often leads you to wives or parents. You just need to follow the breadcrumb trail. There are also free forums where you can search and post genealogical info. It was through one such forum that I found lost cousins and ultimately got to meet them all.

Good luck and happy hunting :)

Sometimes there are community ed. classes on the subject. You might look into that.