Ultimately, intervals need to be committed to memory, both aurally and visually. There are, however, a few tricks to learning this quickly. One such trick is the so-called white-key method.

White-Key Method

The white-key method requires you to memorize all of the intervals found between the white keys on the piano (or simply all of the intervals in the key of C major). Once you've learned these, any interval can be calculated as an alteration of a white-key interval. For example, we can figure out the interval D4-F#4 if we know that the interval D4-F4 is a minor third, and this interval has been made one semitone larger: a major third.

Conveiently, there is a lot of repetition of interval size and quality among white-key intervals. Memorize the most frequent type, and the exceptions.

All of the seconds are major except for two: E-F, and B-C, which are minor.

All of the thirds are minor except for three: C-E, F-A, and G-B, which are major.

All of the fourths are perfect except for one: F-B, which is augmented.

Believe it or not, you now know all of the white-key intervals, as long as you understand the concept of interval inversion. For example, if you know that all seconds are major except for E-F and B-C (which are minor), then you know that all sevenths are minor except for F-E and C-B (which are major).

Once you've mastered the white-key intervals, you can figure out any other interval by adjusting the size accordingly.