The following answers are simplifications, but should be accurate enough. You really don't want the full details yet.
1. Using the example scale above (which is a C scale), all the notes of the scale are on the white keys. A sharp indicates that you go to the black key that is to the right of your white key; a flat indicates that you go to the black key that is to the left of your white key. "Wait", you say, "there's no black key adjacent to the left of my F. How do I play an F-flat?" Well, in that case, F-flat is the same as an E.
2. Think of an entire piano keyboard, stacked vertically so that the lowest note is on the bottom. The 'bass clef' shows about an octave-and-a-half starting at the 3rd floor. The 'treble clef' starts at the 4th floor. On a keyboard, you generally play the notes on the treble clef with your right hand and the bass clef with your left hand.
3. Get a copy of the free
Finale NotePad 2005, which looks really good, and is available for Mac and Windows; once you have that, you can download sample songs and user-submitted songs from their website. I was just reading about it yesterday, haven't tried it yet.