I need to contrast two sets of characters in Shakespeare's "King Lear" in order to better understand the play.
Lear and the fool make a natural pair for a topic like this one. King Lear is a man characterized by nobility of position and also a by a rash temper. The fool represents an opposite character concept, though he possesses a wisdom that far surpasses that owned by Lear.
I'd say the best bit for contrast here would be to contrast the two families that make up most of the play's characters alongside each other. That is, in other words, the Lear family and the Gloucester family.
King Lear's journey runs, roughly, in parallel to Gloucester's. As Lear descends into madness, he becomes wiser, and able to realise more truths about man's condition and about the world:
Thou art the thing itself: unaccommodated man is no more but such a poor bare, forked animal as thou art.
Gloucester too endures a horrible decimation of his body: he is blinded, though, he says, he is better without his eyes:
Lear's two children by one mother turn against him horribly: Regan and Goneril throw him out of their houses and then wage war against him, eventually attempting to have him killed. Lear's child by another mother, Cordelia, is loyal to him until the very end.
Gloucester's bastard child, Edmund, plots against him and his other child, Edgar. Edmund teams up with Regan and Goneril and is, at least partly, responsible for Gloucester being blinded. Edgar, on the other hand, though he runs from his father (he, like Cordelia, is judged harshly and unthinkingly by his father early in the play!) remains loyal throughout too: leading him around once he is blinded.
Hope it helps!