Purchase a book that lists the major and minor scales and discusses their construction, and also keep a book handy that discusses chord construction.
Start by building triads—simple, three-note chords. To build a triad, use the appropriate song key to determine the notes you need. If you are in the key of C, for instance, the notes are C-D-E-F-G-A and B. A simple triad uses the first, third and fifth notes. To find the harmony chord for the note C, you need the notes C-E and G.
Build a major triad on the second note of the C major scale, which is D. To do this, use the D major key signature. D contains two sharps: F sharp and C sharp. This means the notes of D major are D-E-F#-G-A-B and C#. A D major triad (first, third and fifth notes of the D major scale) contains the notes D-F#-A. You can use this formula for any key signature with any note you choose to begin on.
Harmonize a minor chord in the same way. All minor chords are harmonized like major triads, with one simple alteration. Instead of a "natural" third, you "flat" the third of the scale. A C major, as stated earlier, contains the notes C-E- and G. If instead you use the notes C, E flat, and G, you have a C minor chord.
Practice building these major and minor triads in each of the 12 musical keys.