I'm applying the "Subdivision Surface" modifier, and when I change the places of certain vertices and edges, they seem to disappear under the mesh, so I can't see them clearly. Is there something I can enable that will help me see the vertices on top of the mesh?
Try selecting this option:
It will make the effects if the modifier visible in edit mode. the down side is that your edges will be curved.
Selecting limit selection to visible will also make your it visible. The only issue with this is that it can be hard to select the right vertex since all of them, even the ones behind the object are visible.