How do you get out of vault 106 on fallout 3

The same way you came in. I know that sounds like an SA answer, but the easiest way to clear any building, vault, etc. is to start either to the right or left from the entrance and simply continue this pattern until every room is complete. This does get confusing when the buildings are large, have interior rooms, more than one entrance, and places with more than one level, etc, clearing each level and section, as you so desire. Your best bet it to leave a frag mine trail (with the Light Step Perk) so you know where you came from and where you've been).