I've finally figured out movement, heh. Where before I was catching the player input, but not moving the player until the end of the frame, I'm moving the player immediately and waiting a set amount of time until the player can move again. This helps so much, especially as now I can get rid of my previous moveheld boolean, and make my keypress array of integers into a boolean array instead (it had to be integer originally so I knew the key was being held).

With this utterly simplified movement system (whew), adding diagonal movement was a piece of cake. Yay!

Here's a screenshot of the game so far. It's getting there...