Find Familiar says that:
when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it.
So yes, since it's a touch-range spell your familiar can deliver Cure Wounds for you. The player casts it using his action, the familiar delivers it using its reaction.
The moving bit is a little trickier. Find Familiar says that
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn.
This means that the familiar will have to move next to the person you want to heal on its turn to be ready to deliver the spell when you cast it on your turn. It won't be able to move with its reaction, and there isn't a way for you to move it on your turn. (Well, there are some spells you could use to push it around, but not on the same turn as casting Cure Wounds.)
If your turn is before its turn, the best thing to do is probably to Ready your Cure Wounds with a trigger along the lines of "When my familiar is next to Victor the Victim, I will cast Cure Wounds." Then, on your familiar's turn, it moves next to Victor the Victim, triggering your reaction to cast Cure Wounds, and then your familiar uses its reaction to deliver Cure Wounds to Victor. (Using a reaction on your own turn is unusual but perfectly allowable.) The only problem with this strategy is that if your concentration is broken between your turn and your familiar's turn, you will lose the Cure Wounds spell, costing you the action and the spell slot you used to cast it.