I'm incredibly new to Blender, so I apologise if this question has a simple answer.
I'm trying to select vertices and fill them in, but when I do the filled faces seem to overlap with another face.
Any idea where I'm going wrong? Any help is appreciated.
Select all the verts or edges that make up the hole and press the F key. Your hole will be filled in as long as all the verts or edges are part of the same mesh.