Scott - our paper is not very good for learning what a double category is! In particular we need 'pseudo double categories', where composition of horizontal arrows is associative only up to an 'associator' 2-morphism. It's easiest to start with 'strict' double categories.
I suspect the most fun place to start is Eugenia Cheng's video:
* The Catsters, [Double categories](https://www.youtube.com/watch?v=kiCZiSA2W3Q&feature=channel_page).
But she seems to be assuming you've already watched a video on internal categories, which I can't find.