So druids. When they wildshape, does their creature type change? Like, most player characters are humanoid (some are not), do they become beasts when they wildshape? Or both simultaneously?

A Druid's "game statistics" are replaced entirely by the Beast's while Wild Shaped... and then the feature goes on to walk that statement back and provide a few exceptions, so that you end up retaining your class, race, and other features. However, nowhere does it say that the "humanoid" creature type itself is retained.  

Is creature type indeed a game statistic? The introduction to the Monster Manual tells us yes, creature type is a part of statistics. So absent a Wild Shape sentence telling you to retain your creature type, you become a Beast.

Charm Person requires a humanoid target. If you are a Beast, you are not a Humanoid, and Charm Person can't be cast on you. Would I go so far as to say that Wildshaping can end a Charm Person effect which is already taken hold? I'm not sure about that... the spell must target a humanoid, but it doesn't mention ending if they stop being a humanoid... would be a reasonable ruling for a DM to make, but not sure it's required. 

Would use something like Charm Monster since it lists creature.