I've been researching my genealogy for almost a decade at this point. It's a lot of work, but I find it fun and enlightening.
Now, there are really two questions here. First, software. There are many different programs out there, the one that I have the most experience with is Legacy. The free version is quite powerful, even if it's closed source.
The second question that wasn't really asked though is regarding the data. You could have the best genealogy software in the world, but it's useless without the data. I'm afraid that simply adding you, your parents, and your grandparents isn't going to do much for you, no matter the website or software. Part of the problem is that the most useful source of information for ancestry, census records, isn't available for any census later than 1930 in the U.S. due to privacy concerns. Most likely, you'll have to find some great-grandparents (ask your parents or grandparents about them) and from there it's relatively easy to bootstrap your family tree back to the 1850s. Unfortunately, the most complete census records are found on Ancestry.com, which is a pay site of course, but there are so many other records that they have that it can be worth it to pay the monthly fee (I do). In addition to that, you can also piggyback on other peoples' research through their community.
If you don't feel like paying, there are plenty of other sites out there with free data and the number grows constantly. Familysearch.org is probably the best. Rootsweb (owned by Ancestry) is also free and has the World Connect function that will show you family trees (secondary sources). Be persistent. There are plenty of sources out there.