I want to move the vertex in this direction
But it won't let me. I can move it in the X and Z axis but not the Y, like the plane is stuck being 2D and won't enter the third dimension.
It's probably because the scale of that object is all over the place, including the Z scale being zero! In Object mode, apply the scale with Ctrl-A > Scale and you'll find that it works fine.
You deliberately set the scale of Z to zero which in reality does not exist. Set it to a non-zero value.
Alt+H
to reveal everything coz i was thinking it was hiding XD $\endgroup$