Listen to two different notes being sung. One note will have a different texture from the other note. Start with an F# and compare it with a Bb. You will notice that the F# has a twangy sound and resonates powerfully. In contrast, the Bb has a softer sound and isn't as sharp and vibrant as the F#.
Sing all 12 chromatic pitches by first checking them on the piano. Record each pitch individually, and hold the pitch out for at least six beats. Be certain that you are holding each pitch for the same duration. This will avoid the tendency to identify a pitch by its duration.
Pick two pitches to compare that are relatively close to each other. Play them back several times until you are able to identify them by their sound and not by their relative highness or lowness. Listen casually and try to get a feel for the individual qualities of the notes.
Continue adding one pitch to your exercise routine until you are able to identify all 12 pitches. This could take more than a year to learn. The brain does not learn this information easily.