Animation in particular is something I would like to know in detail. I checked out a book and I feel like there's a lot of groundwork I need to familiarize myself with before getting there (lot of math).
Animation and graphics is where math is KEY. Matrix and vector arithmetics must become your friends, same with trigonometry, linear algebra and quaternions. More advanced features in gfx/physics will require you to dive into new math areas but this is something you'll soon discover once you start research on a given topic.
At what point do you call the application an 'engine', as in what exactly are the bare essentials. Is that when you have a way to define game objects, have some physics system set up, and can have scripts running?