Is there a way to Prevent Meshes from going through one another? I want to treat my individual meshes as solid objects where the other mesh is unable to pass through them. If I were to push the mesh into them for instance, it would stop at the border of the mesh instead of continuing through.

Is there a way to make these meshes have these "barriers" so to speak?
If you're referring to while you're editing the scene, this isn't really possible. For animations, you should research Blender's Physics Simulation capabilities.