Story is independent aside from the passing where I believe our characters from L4D interact with L4D2 characters. I haven't played the passing yet. I would play L4D just because, but no you can play L4D2 right away without missing any story.
Last time I checked the 'story' for L4D(2) was as important as CS and TF2.
In other words, hardly a determining factor. The scribble on the walls are about the only hints of whats going on but they're not relevant to each other L4D2 game.
L4D1 has different characters and maps from 2 so its worth checking them out if you haven't.
Here is a mod that gives you all the graphical changes and effects from 2 and puts them in 1.
I tried it out and it was quite fun, brought new life into the old one. Only thing is , is that all the characters had the voices of the people from the second one, which was kind of a bummer because the characters were so much better designed personality wise in the first one.
I found Left 4 Dead 2 to be a much better game from the personalities of the characters to the variety in the campaigns. You can get in on a group buy for $15 which is too good a price to pass up. Don't bother with the first game.