There are two music notation programs by which all notation programs are measured: Finale and Sibelius. Of course, these two programs can do virtually anything imaginable in music notation, so no free program is going to compare. But, for students just entering the world of composition, there are a number of free notation programs that you can use until you find yourself frustrated by its limitations.
MusetteMusic offers a free downloadable program.
MuseScore posts their handbook here.
Lilypond offers a free downloadable program. I’m not impressed with their website, so it wouldn’t be my first choice.
Based on its screen shots, I think that MuseScore looks the most user friendly.
Because I am a Finale user, I would be comfortable recommending their entry level product, Finale Notepad, which you can download for only $9.95. 
But, you may be just as happy starting with one that is free.
If you have any experience with these and want to leave a comment about them, please feel free!
For a comparison of MuseScore and Finale NotePad see http://musescore.org/en/comparing-musescore-and-finale-notepad-feature-feature
Although I have used PrintMusic and the full version of Finale for years, I wouldn’t recommend Finale NotePad.
I have set things with MuseScore that others have sworn at Finale for its inability to do (certain types of accidentals, for example). The free MuseScore can set unlimited numbers of staffs in a single system (at least, up to the limit of the paper), change key and time signature any time you want (which Finale Notepad can’t do), and has a really simple keyboard entry system (just type the letter of the note you want, and it will guess which C you want next). Or you can use a MIDI keyboard.
Lilypond, on the other hand, is a page description language. The learning curve is very steep, like that of LaTeX (the equivalent for text processing). But once you’ve mastered it, you can typeset literally anything, and it will be printed correctly. (The website has examples of scores that Finale sets incorrectly, but Lilypond gets right.)
Thanks for the feedback. See my updated post for updated recommendations!
Thanks for this link, David. See my updated post for an updated recommendation!