Thismay sound kinda lame:
Start out maybe 5 years before the great war and introduce us to the game,like 3 and NV did...
And maybe show cinematic footage of news videos of the growing threat of Nuclear war and ads for the Vaults.
Have the years pass to weeks before the bombs hit and you and your family get accepted into a VAULT,just before the bombs fall.
As the door seals and the bombs fall show another cinematic of that and maybe a presidental conferance of some sort.
Then a few more years pass and do a little of life inside the vault, making friends and the like. Then one day the door just opens and as people leave and die from radiation and the like, you are thrust into the after-math of the war.
The city or area would be nice to see Miami,New York,Chicago,Detroit, Japan,Europe, or Canada, or back in California, or even texas.
From there its survial and trying to learn to adapt to the Wasteland.
Not very descriptive but I always hoped for a taste of the pre-war life.