I'm brand new to 5e and want to play a cleric. When I last played over 30 years ago, the cure wounds spell could either heal a PC or damage undead. Is that still the case?
Nope, this is no longer the case!
Spells (and all features etc) do only what they say they do. Cure wounds very specifically says:
This spell has no effect on undead or constructs
So in fact it does nothing to undead at all. Neither heal nor harm.
Specifically, there is no concept of healing "positive energy" and harmful "negative energy", and so no need for special rules concerning their interaction between living and undead creatures.
5th edition generally has a rule system with far less "interactions" between rule elements like this, with the result that it is far simpler to know what a spell or any other feature does: it does what it says it does. No more, no less.
This can be very confusing if you are used to older versions (I know, I made several mistakes myself when switching over)!