Practicing harmony while at the piano can help you understand what harmony really is. If you know that a C major chord is made up of the notes C, E, G, or 1-3-5, you can begin to understand and practice harmony. Play the notes on the piano, one at a time, and begin to sing them. Play a C note on the piano, and sing an E note at the same time. This is a basic harmony. Try the same thing with the G note.
Sing along with a recording of a song you know well. Try to sing a slightly higher or lower pitch than the singer on the recording. If it sounds good, you are harmonizing with the singer. Unless you have very good pitch, it may be difficult to learn what notes you are singing this way. Still, it helps to give a basic idea of what harmony really is. Do this with different chords, and think about the relationship between the notes.
Practicing harmony with one or more singers can be a great way to learn and hone your skills. Take your group to the piano, and play a chord one note at a time. Each singer can then sing one note of the chord. Change the note you sing within the chord, and change the chord you are singing as well. This will help you to commit the notes to memory.
Recording yourself while harmonizing with the piano or other singers can greatly improve your skills. People rarely sound like they think they do, and they don't usually sound better than they think. Don't be discouraged when you hear your first recordings. Harmony is not an easy thing to learn, and it is a very difficult thing to master. Practice different notes and chords until you are comfortable with them, and then move on to others.
Before you open your mouth to sing, make sure you can hear the note in your head. If you have no idea what it sounds like, how can you sing it? Until you can imagine what the note is going to sound like, it is going to be hard to sing the correct pitch consistently. Practicing at the piano and recording yourself will help quite a bit with this, as well.