I'm wondering how I would render a cube with a different PNG Texture on each side, and I'd like to stay away from other Third-Party Libraries other than LWJGL.
As of LWJGL3 bindings for the STB library have been included. This is a small single-file utility library that does things like loading textures, sound files or fonts.
For rendering a cube with a different texture on each side you have two options:
Introductory tutorial on how to use textures. (It is in C++, but should be easy to convert to Java.)