Translate the C major scale into numbers. The C major scale, like all major scales, has eight notes: C-D-E-F-G-A-B-C. When translated into numbers, it becomes 1-2-3-4-5-6-7-8.
Learn the formula for major scales. These scales are made up of steps -- the distance between two notes -- and these steps come in two kinds, whole (W) and half (H). For example, going from C up to D is a whole step, but going from C to C-sharp is a half-step. The major scale formula is W-W-H-W-W-W-H; practice hearing the difference between half and whole steps.
Use the major scale formula to build all the major scales. Start with the note of the scale you want, then apply the W-W-H-W-W-W-H formula. If you start with G, then the notes of the scale are G-A-B-C-D-E-F#-G. Keep in mind that even though you start with different notes, the sequence is always 1-2-3-4-5-6-7-8.
Start to build chords using what you know about major scales and the numbers. To play a basic major chord, play the 1, 3, and 5 notes at the same time. In C major, 1, 3 and 5 are C, E, and G; in G major, 1, 3, and 5 are G, B, and D. Practice these chords in as many keys as you can.
Practice your major-key chords again, but this time lower the 3 note by a half-step. For example, instead of playing C-E-G with 1-3-5, play C-E-flat-G. Now it is a C minor triad instead of a C major triad. Go through the major chords you have played and change them into minor chords.
Apply the Nashville Number System. Major and minor chords can be constructed on each note of the scale, and these are numbered the same way: 1-2-3-4-5-6-7-8. The chords at 1, 4, 5 are major, while the chords at 2, 3, and 6 are minor. The chord at 7 is a "diminished" chord, and 8 is the same as 1. Many popular songs just use the 1, 4, and 5 chords; a typical Nashville chart looks like this: 4 / 4 / 1 / 1/ 5 / 5/ 1/1. In the key of C, that works out to F / F / C / C / G / G / C / C.
Practice charting out songs with the Nashville Number System. Once you have figured out the numbers for each chord in the song, play it through. Then try it in a different key, using the same numbers. Start with a new tonic note -- G instead of C, for example -- and use the numbers to play the song again.