Now this doesn't have to be an expensive process. Yes, you do have to pay to use http://www.ancestry.com but you can find so much information that is free.
https://familysearch.org/register is the website I used to create a free tree.
You will have to create an account with some standard information. Once you get your email confirmed you will be able to start your tree.
You can also use Myheritage.com, it is free up to 250 people, which happens fast. It is also like familysearch.org where it will give you hints on who could be next in your line. It also has a program you can download to keep all the information you find saved in your computer, offline. I have this but I don't use it as much as I did at first.
First you will need to enter in your parents information. Date of birth and place of birth. More than likely your parents are not old enough to have a bunch of records yet. Due to privacy laws, records have to be so many years old before you can view them.
You will hopefully be able to move on with information that you have for your grandparents. Place of birth and date. This is where the familysearch.org will start to fill in information for you and this is the point where you need to make sure that the information you are finding is accurate.
It will be tempting to just go with whatever pops up and think that it looks correct but you should always triple check. There should be a few record sources that can back up dates and places and that's how you will know that it's correct. If you can't find records than now is the time to start searching.
All of the information you find will automatically be saved in the familysearch.org site so you can come back to it whenever you want.
You can also be looking for information on family that has passed by using findagrave.com. This site has a been a wonderful tool. Some people make it there mission to create profiles for people in cemeteries around them, so if you are looking for proof that a person has passed I would search there. I have found so many relatives and been able to go to the sites to see for myself.
I hope that I have helped get you ready to create your own family tree and if you have any questions feel free to ask and I will help to the best of my ability or if you have a wonderful trick you use please share.