Hi there, following less 23 I had problems in selecting two vertices there were perfectly overlapping.
I think that these very trivial and basic operations are the less intuitive in Blender, if you come from 2D apps and they can really make you stuck.
The solution was somewhat not too complex:
right click to select one vertice
press and keep pressed shift key
right click in the same place as before to select the other vertice
That is decent solution to dealing with overlapping vertices.
You can press A in the 3D view window to select everything while in Edit Mode and then Remove Doubles, but you might not want to do everything.
I recommend that you enter Wireframe mode (Shortcut Z) and use either the Border Select (Shortcut B) or Circle Select (Shortcut C). I prefer the Circle select much more. You can press Right-Click to cancel the tools. And you’ll learn how to use them more in the future.