Learn the notes, and their positions on the stave, or staff. For the treble clef, the bottom line is E, the next line up is G, then B, D and F. These are easily remembered by the mnemonic "Every Good Boy Deserves Fun." The spaces, from the bottom up, are F, A, C and E.
Practice naming notes. Pick a note at random out of a piece of music, and call it by its letter name. Make sure you're reading in the treble clef. It's essential to embed the notes and their positions in your memory, so take your time and learn them well.
Teach yourself how the notes sound. Look at the music for a well known tune, and play each note in turn on the piano, whilst singing along. Notice how the higher notes on the stave are higher in pitch.
Learn the different intervals and how they sound. Practice singing alternating thirds, fourths, fifths, sixths and octaves, always looking at the notes you are singing written on the stave. You will learn to associate what you hear with what you see, so use the piano to make sure you're getting it right and in tune.
Try reading a new tune that is unfamiliar to you. Break it down into short phrases, and sing each one before checking if you were right on the piano. If you weren't right, listen to the notes on the piano, then try again, still reading the notes from the page. Watch out for key signatures and accidentals. A sharp sign means the note is a half-step higher than usual, while a flat sign denotes a pitch a half-step lower.
Start with the bass clef, once you're getting comfortable with the treble clef. On the bass clef, the lines from bottom to top are G, B, D, F and A ("Great Big Dogs From Africa"), while the spaces are A, C, E and G ("All Cows Eat Grass").