Discrete-time Fourier transform
It's great that you are interested in signal processing at that early stage of your educational path. 

The best path to get there is to read some introduction books on the topic. There are a lot of good and free online resources to get you started. [Note to the esteemed editor: good introduction books may a really good topic for a "sticky"].
I sometimes use 


http://www.dsprelated.com/dspbooks
http://www.dspguide.com/pdfbook


One of the most important mathematical concepts you will need to get your arms around are “complex” numbers”. It’s clearly a misnomer since it’s really not that complicated and clearly makes nearly all engineering math a lot simpler. Another great free resource for all things math related is http://www.khanacademy.org and in this case specifically 
http://www.khanacademy.org/video/complex-numbers--part-1?topic=core-algebra

Back to your first question: There are actually four different flavors of the Fourier Transform: Fourier Series (most likely to show up in high school), Fourier Transform, Discrete Fourier Transform and Discrete Fourier Series. All of them of them use a combination of both sine and cosine (or a complex exponential, which is essentially the same thing). You will need both. 

Let’s say you calculate the sine and cosine Fourier coefficients of an input sine wave. (Under certain conditions) you’ll find that all Fourier coefficients will be zero except for one cosine and one sine coefficient. However, depending on the phase of the input sine wave you these two numbers will move around. You may get [0.707 0.707], or [1 0], or [0 -1], or [-0.866 0.5] etc. You will see that the sum of the squares of those two numbers will always be 1, but the actual values depend on the phase of the input sine wave.

If you want to deep dive, try this: 
http://www.dsprelated.com/dspbooks/mdft/