One of the basic skills you should work on developing through ear training is recognition of intervals. There are twelve pitches in the Western chromatic scale, which means that there are twelve possible intervals, or melodic jumps from pitch to pitch. Each interval has a distinctive sound, whether it's used melodically (one note played after the other) or harmonically (two notes played at the same time).
A good way to identify melodic intervals by sound is to learn to associate them with particular songs that use them. For example, if you can remember what "Twinkle, Twinkle, Little Star," sounds like, you can remember that the first interval jump in that tune is an interval of a perfect fifth.
Use a piano to help develop your sense of intervals, and, if possible, a partner. Quiz each other on different intervals by taking turns playing them at the piano, not allowing the other to see the keyboard. If you don't have a partner to work with, sit at the piano and try to reproduce a particular interval by singing it, then playing it on the piano to see if you were right.
Next, work on learning to identify harmonic intervals. With these, you can't use a song or trick, but each interval has a distinctive quality that you can learn to recognize by listening to them often.
Intonation is a performance-based aspect of ear training that no musician should ignore. Use an electronic tuner to help you practice (available in any instrumental music store). Buy a machine that can play pitches, as well as show you a gauge display that indicates whether you're in tune as you play or sing. Practice matching the played pitches by ear, then use the gauge to make sure you're getting it right.
Learning to identify the chords in a piece of music by ear is a useful skill for any musician, but especially for composers and improvisatory musicians.
In order to improve your abilities in this area, you must be able to work with a partner who can play chords, or use an audio track. When first starting out, work on learning to identify chords by their quality. A chord's quality is its general color of sound, determined by the stacked harmonic intervals that make it up (such as major, minor, augmented, or diminished chords, though there are others). Each different combination of intervals makes for a different quality.
Practice identifying the quality of single chords on their own, then work on identifying chords in the context of music. Some good music to use for this purpose is four-part church hymns and the chorales of J.S. Bach.