Have to agree about the video. What he's saying is the right idea (as in driving from your legs into the bridle), but the execution is a bit lacking. He's hanging all over that poor horse's face
Anyway! Back to the questions.
Everyone else has covered the basics, but no one's really mentioned bridging. This isn't really necessary unless you're planning on showing, in which case this would be the "correct" way to hold the reins while riding two handed (assuming your horse is still young enough to be ridden in a snaffle).
So when bridging the reins, you overlap the two split reins and pick them up holding them like you would riding in regular english reins. The right rein will still fall over the left side and vice-versa, but you will be holding the end of each rein in the opposite hand creating a "bridge" from one thumb to the other. I'm a fan of visual aids so I've attached a couple pictures below so you can see what I'm talking about.
When riding with bridged reins it is important that you have enough of a bridge (ideally 8-12 inches of rein) between your hands so that they can work independently of each other. If you have to little rein between your hands, you won't be able to cue the horse with one hand without dragging the other along for the ride.
As for taking up on the reins, it works pretty much the same as it does in english riding. You just walk your fingers down the rein until you've got what you need.
Hope that helps!