Just go to settings > display and gestures, then turn on Auto Rotate. That will automatically change orientation of videos and apps when you turn your phone sideways. In YouTube, it also acts to change to full screen without having to tap the expand button.
There's also a quick setting for this. Pull the notification bar down with two fingers (or with one and then tap the group of squares in the upper right corner) to access the various toggles. There may be a tile for auto rotate there. If not, tap the edit button (upper right, looks like a piece of paper with a pencil over it). There, you can reorder and select which tiles you want visible or hidden.