Note: This is obviously not official. Just some thoughts from my end.
Game will begin with a cinematic explaining the overall tone of the game, starring Ron Perlman's voice.
Once cimenatic ends game will begin with pre-war elements like shown in the trailer, either with gameplay segments where you for example have to run-down the streets and walk up the hill to the Vault entrance or only a video.
As soon as the pre-war segment ends you, the player, will wake up out of cryostasis only to see that the vault has been completely empty. No human to be seen but there will be life down there....(a) robot(s)! You as the player will interact and speak with them/him and he'll probably tell you about why you're the only one left down there. There will probably be some kind of Tutorial for V.A.T.S. and other stuff just like in Fallout 3 with the Vault 101 part. Also there is going to be some kind of explanation and part where you can decide how your character looks like. Either male or female, like you've experienced it in previous games.
Then you will get info that makes you want to walk out of the vault, you'll go to the entrance, open the giant Vault Door and probably think of another cave entrance with light in the near distance. But nope. instead you will be greated with another empty and dark room. The elevator shaft of Vault 111. There you'll have to walk on its platform and find the switch which will bring you to the surface of the real world. Welcome to the world of Fallout 4.
You will begin to search for clues in the near houses down at the river and scavange some supplies which should help you get near to Boston. in one of those houses you will find some super cool looking Laser pistol which wiil help you alot on your way. Once you make it to the entrance of the city you'll be greated by some gritty-looking mercants where you make the decision on selling your laser pistol for some caps. Coming closer to the city centre you'll notice a huge building with red neon-lights hanging above its entrance. As the curious vault dweller you are you make your step inside.
tl;dr: you sell all your stuff on android prostitues and it will be amazing and immersive.