The main thing is to unwind, not cut, the old strings off, by reversing the pegs, so that the pegs come up to receive the new strings and new windings. This prevents you from screwing the pegs down too deep-- bad things happen.
Me, I take the coward's way out-- I let the repair shop do it when it's the whole harp. They also check out all the chord bars, springs, and spring holders. On a 21-chord model, the cheap plastic teeth that hold the springs and chord bars are extremely fragile-- not only can there be one tooth broken off or about to give way, but I say let the pro break and replace them just opening up the damn top or putting it all back together!
They polish up everything while the harp is sitting there naked, so to speak.
(There are kits from Oscar Schmidt that replace the whole top assembly too, if this is an old beat-up model.)
I oil my strings, BTW, to slow down oxidization, and it seems to improve tone brightness and sweetness too.
But never mind all that. I wanna know-- are you electric or not (do it!!!) and do you play blues on it? (Do that too!) If so I would love to correspond, or hear more here in this thread about how and what you play. (Tell us more about your harp too.)