As BoomCheese suggested, I'm quite certain that all we will be doing inside the Vault once we wake up is leaving it. That is of course, once we've found the jumpsuit, the 10mm pistol, and the Pip-Boy. Anything else seen in the PCs inventory would have been for demonstration purposes only, otherwise he would have used a more advanced weapon to kill the mole rats outside the Red Rocket garage.
The events that transpire will be to do with having to go into cryogenic pods to prevent you and your family from being killed by overpressure. And the twist, from what I can tell, will be that you have to watch your family 'drown' in the pod opposite you due to the requirement for it to be filled with cryogenic fluid as part of the freezing process. The extreme high pressure would prevent the fluid from solidifying into ice which would kill anyone inside. You would then go through the exact same process yourself, time would pass, and then when you wake up, you find yourself choking on the fluid as it drains from the pod. Military training, from either the US Navy or the Marines depending on choice of protagonist, tells us that if we don't remove our soaking wet and freezing cold clothing straight away, we could die of hypothermia. And so we find and put on the Vault 111 jumpsuit that we see the PC wearing in the footage. This would even explain why we see him catching his breath as he leaves the vault. As for the family, the pod containing Shaun and the spouse would have been removed several years prior by the Institute, who are the main focus of the game, especially considering the original codename was actually "Institute". Again, when the PC speaks to Codsworth, he seems more than a little confused as to the whereabouts of his family.
This is, of course, all speculation. But it does kind of fit what we see before and after.
EDIT: the fridge in the player house even has a viewing window near the top of the door. I have a feeling this will be reflected in the design of the pods.