In Belfast/other parts of Northern Ireland we sometimes greet people by say, "'Bout ye!" or "Whata 'bout ye!" = How are you?

You can use that if you want!

For colloquialism:  'hanging out'

I taught my kids "Howdy!", "TGIF!", "chilling," (among others) and they seem to remember those three in particular.

Trying to find timeless and non-regional things may prove too difficult. Honestly there's a lot that we say that we assume everyone else does, but that often isn't the case.

For example, in the States we'd say "She cracked up" and it means she laughed. In the N. Ireland if you say the same thing, it means she got angry. In the states if someone "takes a crack at it" they're giving it a try.
Likewise in the UK/N.Ireland, if someone is "cracking" or "cracked" they're crazy. But then again if you ask "what's the craic/crack?" You're asking what's new.

Without travelling to N. Ireland and learning all of this, I would have just assumed that "cracked up" means "to laugh" everywhere.

I'm sorry I can't really offer any actual suggestions. I just can't think of anything that's guaranteed to be region-neutral, at least for now. Maybe it's best to just teach what you know with the preface that this stuff is only used in whatever country you hail from?