One of the best ways to organize your family tree is to enter all of the details into genealogy software. Once you have more than 20 or 30 people in your family tree, keeping track of them on paper is not nearly as efficient as using software. The software gives you easy access to both the overall tree, and all of the branches and little twigs.
There are various programs available but there is no need to spend any money. One program that has everything you need for a PC is provided free by the LDS Church. It is called PAF (Personal Ancestral File). To download the current version, go here. Follow the instructions to install the program on your computer. As you use it, the program will automatically create a backup copy (it will ask you to update as you change the file) so if you lose the file or make mistakes you can reinstall your backup without losing your data. PAF allows you to view individual people, families, family lines, and so on. You can add pictures and other information about your relatives, and you can create and download reports that are compatible with MS Word etc. It is easy to use, and there is loads of free advice available if you have questions!
PAF is not available for Macs - you will have to do a bit of research to find something that you like - try this link for more information.
Sites like Genes reunited and Ancestry also allow you to store your family tree data. However, using software such as PAF gives you more control over your data and is free of all the ads that will accompany the commercial websites. As well, the latter tend to be a lot clunkier to use - more clicking and page-loading needed.