The first thing you need when starting a genealogy project is a way to collect all of the data that you find. If you are not organized, you are going to get frustrated very quickly. With that in mind, you must devise a system of record keeping.
Until recent times, most genealogy record keeping was done on paper and stored in files. You could get large charts from the local LDS library and had to write everything in pencil so you could make changes. I literally have a chart that is the size of my living room, not that I use it anymore. I keep it as a reminder of the way things used to be.
Maintaining paper records is always a good idea but you have to come up with an adequate filing system. I always make folders, going by last name, first name of the husband and keep records for himself, his wife and any unmarried children in that file. I then put all of my folders in a file cabinet, in alphabetical order by that last name.
Having a bunch of papers does not constitute a family tree. To see the fruits of your labor, you need to create a coherent accounting for the information found on each document. In the older, manual method you would use "family group sheets" to document the names and dates associated with each person in a given family. These are simple paper forms that most local libraries, at least those with a genealogy focus, can give you. You can also find online versions that you can print off for free by using this link: Ancestry free genealogy forms
The modern way of organizing genealogy data would be a computer-based database. There are many programs on the market, ranging from one hundred dollars down to free. More expensive does not always mean better though...keep that in mind. There are also web-based programs that you can use, again some are free and some charge a fee per month/quarter/year. A few of the most popular online programs would be Ancestry, My Heritage, Geni and Tribal Pages.
I tend to stay away from the online database programs, for several reasons. You sometimes lose control of what you put online. Most sites allow you to keep your work private but you really have no way of knowing exactly how safe and secure their site really is. If they had a computer crash or went out of business tomorrow, what would happen to all of your research? I am probably being a little paranoid but I like to hold my hand closer to my chest. I would recommend having a program on your computer and then uploading that information to a web-based program, if you want it on the web at all.
As for computer-based programs, there are lots of choices out there. I have always used Family Tree Maker for my own research. The most recent versions have had some growing pains but the program is still the most used in the industry. Another popular option would be RootsMagic. Reunion is popular with those who use a Mac/Apple computer. There are also a few free programs that you can download, the most popular being Personal Ancestral File or PAF. It is made available by the LDS church. I have not used their program personally but it is widely utilized throughout the world.
Follow some of these links and check out each program before making a decision. A good site for product reviews is GenSoft Reviews. There you can read hundreds of personal reviews of over 400 different genealogy programs.
After deciding on the right program for you, take the plunge and get one. Most can be obtained online but you can get the more popular programs at Target, Wal-Mart, Best Buy, etc. Try to get the most recent version, not something that has been sitting on the shelf for three or four years. You can also pick them up at eBay but only buy those that are "new in the box."
When you get the program in your hands, install it on your computer and start playing around with the features. Get bold and enter what you know about yourself and your immediate family. Once you enter your first name and date, you are officially initiated into the genealogy hobby. I warn you now that it can be extremely addictive!