Harmonise eh? First thing first, do you have a keyboard or a piano at home? If so it’s pretty easy to do so.
Let’s take the note C. If you want to harmonise in C Major just sing the note which is four semitones above it (each semitone is each key, doesn’t matter if it’s black or white [ooh ooh ooh]). In this case, the note to sing is E. You can also harmonise by singing the key seven semitones above it. In this case that’s the G key.
Here, take a look at this C Major keyboard illustration for reference.
I believe that this is a good starting point. Just practice with the basic notes and keys and play around with them with your voice. It would be good to also practice along with a musically-inclined friend. I sometimes just sing with a friend of mine and since we’re both quite solid in our musical foundation (and also because we have relative pitch [mine’s passable, meh]). Note: Relative pitch is, extremely simplistically speaking, your ability to harmonise (correct me if I’m wrong Flutherites)
Having a musical instrument at home would really help, in my opinion, although I’ve never really tried to train myself to harmonise. I just like to sing along, and occasionally harmonise when it feels right.