Why isn't the vertex moving the face of the object? Hi I'm just learning how to use Blender and got stuck trying to add vertex points to an object that aren't changing the shape of the object.  

I know if I press F I can add a new face from the selected vertices.
But is there a more efficient way of doing this? When I add a new face I now have to delete the one under it.  Is there a way to add vertices that are already connected to the face of the object.  (sorry if I'm not using some of these terms correct)

Here's a screenshot of the vertex I added by subdividing two edges, connecting those two vertices, and subdividing that new edge.
You almost got it right.
Just use J instead to connect the vertices.

1. Connecting vertices across a face
Vertex > Connect vertex path
Shortcut: J



2. Connecting free floating vertices
Vertex > New edge/Face from vertices
Shortcut: F