I haven't played anything pre-VII, so I'm getting ready to be in the same boat playing through them. Currently I'm enjoying XVI.
If you liked the remake, I would say start with VII. The original is the complete game whereas the remake is obviously only part 1 so far. However, you'll have a better appreciation if you play the original, because in playing the remake you'll notice that some things aren't quite what they should be. And Aeris seems to have some awareness of this.
As far as skipping, you've already noticed that the games aren't connected, so that's up to your preference. If you liked VIIR, go for XV. XIII is an acquired taste is all I can say for that. Though I would put in a vote for IX as well.
But the stories in most of these games are truly something to be experienced one way or another.