It is a free site, and has many vital records and census records. You can also go to the genweb sites, which often have good data. Just google US genweb + the state you want. Also, public libraries sometimes have genealogy sites with records. If you are not in your own city, you can still go to a local public library and see if they have free access to Ancestry on their computers. If you have a library card for your home community, you can check and see if they offer access to Heritage Quest, which has census records. I can access that site for free from home, with my library card.
I would recommend finding your family in the census first. The 1940 census is the most recent one to be released, so hopefully your Dad was born by then. Once you find him in the census, you can work back to your Grandparents.
Start by writing down everything you know about your family, then list what you need to find out, such as birthdates and places, and marriage dates, etc. Make sure you verify with documents when you can - don't count on other's genealogy to be accurate. Also, you are not going to find your genealogy laid out all nice for you, you will just have to go step by step.
Use these message boards to get information and maybe find other family members. Post on the county where the person lived. If you don't know a county that a city is in, just google it.
Hope that helps. Let us know if you have other questions.