Let's start with the fact that the OP is only partially right. "To get with so." does not exclusively mean "to date so." It can also mean, in a loose sense, to consult so., to hang out with so., to partner up with so., etc., etc. As always, context is everything.
So how do young people (Americans, in my case) express the idea that they're dating someone? That may depend a lot on how old they are, were they live, which ethnic background they subscribe to, etc. I think most adolescents still "date." Others might just "hook up" (which always seems more like a very short-term situation, but not necessarily so), or they're seeing each other, or they go together, or they're going steady, or they're an item, or in a relationship. These days, however, I suspect they mostly text each other.