This is a common problem that many new users have, so don't worry. It really impacts ex-CorelDRAW users as CorelDRAW is MUCH easier with anchor point editing than AI.
The problem is that the anchor points are very small and hard to see on high resolution screens. Basically, after you draw the shape, the anchor points are selected. So if you immediately grab one with the Direct Selection tool and drag it, it will drag ALL anchor points that are selected...therefore the entire path. If you click off the path to deselect it, then go back and select an anchor point, that might be the easiest thing for you. Or, you can press Q to drag around the anchor points you want, then press A to move them. There are many different methods.
Turning on Smart Guides might help you (Ctrl+U) which allows you to hover over a path with the Direct Selection tool and see where your anchor points are when you're over them. Otherwise, you have to carefully hover and wait until the hollow square appears next to the cursor.
Also, make sure you have "Show Edges" enabled (Ctrl+H). This also confuses a lot of people.
Not sure about your last sentence...unless your path has no stroke color applied. This will cause it to display while you're editing it, but then "disappear" when you are finished and it is deselected. Is this what is happening? If so, simply hit the "d" key while editing the path and it will automatically set it to the default attributes (white fill and black stroke color)...you should then be able to see it.