So this time around I used Krita, a free open-source art program, applying some tips from this post I found. (You might find Pixilart easier to handle since it's designed for this sort of thing, as I used a similar program for the previous game.)
Basically, I started with a 96x96 pixel art, then resized it to 384x384 so it displays better. I don't remember why I chose those dimensions exactly, but that's how it goes sometimes.