There are so many things that you can do to an object that they are separated into different modes. The default is the Transform mode, where you can drag stuff around. The others allow for things like adjusting Anchor Point, Four Corner, Crop, etc. The last mode on any object will be specific to that object type. For example, a Light Rays filter has a centerpoint control, a Gradient generator has endpoint controls, and shapes/masks have points.
The reason why your shape points disappear is because selecting a non-shape object will switch the control mode back to Transform, thus hiding the points. To get your points back, select the shape, right-click on it in the Canvas (or Ctrl-click), then choose 'Edit Points' from the first grouping in the contextual menu. You can also repeatedly press the Tab-key to cycle through the different modes. An even better shortcut: double-clicking on shape objects (in the Canvas) will jump you straight to the points-editing mode.
Sorry for the long explanation, but I thought it might make things a little less mysterious. It's much easier to explain in person :)