Make sure you are using the Subselect tool (the white arrow), not the Select tool (the black arrow.)
(1) select the vector object (all of the points will be hollow or white)
(2) use the sub-select tool and select the one point you want to move (it will become filled or blue)
(3) use the seb-select tool or the arrow keys on your keyboard to move the selected point
You can also use the sub-select tool to select multiple points (by SHIFT+click) and move all of them at the same time.