Thank you for the link. Although it didn't have any info on Goom, there's definitely some good stuff on there.
I took a look at the compiled Goom file size, and it's already much bigger then the Flash size on that board (apologies to everyone for posting my question before thinking to check that).
Luckily, I was able to find a few small STM32F407 boards available online, and ordered one.
The only thing is, I'm not sure where to find schematics to build an Audio out jack like the one on the discovery board. I suppose I'll post a new thread for that when it arrives (if I don't find schematics before then).