... by talking to your living relatives. I wish now that I had gotten my grandmother and parents to talk more. I never even knew three of my grandparents. Consider recording your conversations with the important people in your life. (This is a great project for a child, by the way.) Digital tape recorders are inexpensive. Find any family Bibles, which may contain names and birth dates and dates of death. These are treasures; if you find them talk with the family about preserving them. You may find other family members who have already been researching. Most genealogists are very generous and love to share. Etiquette point: do not post information about living family members on the 'net, and give credit due to the person whose research you are using.
You are wise to ask questions before you start. You can avoid the common newbie mistakes. (Personal experience talking. Genealogy is addictive. I never thought my family would take me where it has or that I would get hooked on it!)
Here we go.
Be prepared for a ton of information. You will want to organize it. There are inexpensive programs --- some are even free --- available. I use RootsMagic; others here will tell you what they are using. You can evaluate several and see which one you like best. These programs will all have a feature that lets you document the source of each fact you find.
Use it!
Document! Document! Document! You will save yourself tons of time and aggravation if you do this from the beginning. The documentation can be as simple as "Conversation with Jane Doe Jones 19 Jul 2010" or as complicated as a book citation for a research paper. Your genealogy program will save references, so if you need to cite the same source over and over --- and you will --- it will be easy. Once you start recording facts and names
back up regularly! Thumb drives make it easy. Know how your program stores photos, so you will know the best way to organize photo files and copies of documents that you scan in.
Once you talk to the family, you will have an idea where you need to go next. Genealolgists like to use primary sources for a fact where possible, for example marriage certificates. Since birth certificates and death certificates were not issued until relatively recently in the USA, dates of birth and death many times come from family sources, like Bibles, and church records. If most of your research is in Canada, things might be a little different. My research has not carried me there. And genealogists spend a lot of time in courthouses looking at estate records and land deeds. This may sound awfully dry and scholarly, but we do get caught up in the scent of the hunt and love to come across the document that proves who someone's parents were. Then there are graveyards. Everyone ends up in the cemetery, photographing headstones. Just beware: this is a source for birth and death dates, but just because it is written in stone does not make it true! I have found stones that I know were incorrect, including one several times great grandfather who was alive in the federal US census after his headstone says he died. I found his obituary and got the actual date from that.
Libraries frequently have entire collections dedicated to genealogy. It is likely you will eventually connect your ancestors to lines that have been extensively researched and you may find books on those families. My local library has records from virtually all of the US states, including marriages, newspaper abstracts, and books on local history. Then there are state history and federal archives. I suspect that there are provincial archives in Canada, too. You will find entire newspaper, too, commonly on microfilm. There may be a research librarian who will be happy to help you find stuff.
You are likely to find much information right here on the internet. Google names and dates and you may find someone else researching your family.
Then there is
Ancestry.com -- Browser Upgrade. Ancestry has US and international records, with an enormous data base that is growing daily. It is a subscription service, but there is a two week free trial. I would suggest getting your basic information, then sign up for the free trial when you are going to be able to devote time to exploring it. There is a library version, but you may have to physically go to the library to access it. Ancestry has a message board with boards for specific surnames and geographical areas. Ancestry also has family trees. Individuals post their trees and you can search them for your own family members.
There are many errors in these trees. People then copy the erroneous info into other trees and it is there forever! Check the documentation before you accept anyone else's "facts." See other threads in this forum about that problem.
One of the first things you will want to check out is census records. These are probably different from US records, so I will defer to Canadian posters to tell you what to expect. All the US census records are there except for 1890, which was almost entirely destroyed by fire.
There are many other databases and message boards, too.
Be aware that standardized spelling of names is a relatively new concept, which can mean that you have to search under different spellings for names. The search engines try to do this, but it is not very efficient. Old, handwritten records can be hard to read, and transcribers frequently come up with some really strange names.
Wecome to a really satisfying hobby. I have only been doing it about three years and I have met some of the nicest people you would ever want to know through online message boards.
Edited to add: I did not see Slim's post until after I posted mine. Hello to another addict!