I picked Fallout 2.
Maybe it's just because it's what I played first.
Tempted by all the hype in this very thread, I gave Fallout 1 another playthrough. And I was indeed able to appreciate some things even more than before (especially how well thought out many of the random encounters were). I also had to once again acknowledge that the plot is very good.
The engine isn't entirely bad, either. It lacks the terrible corpse bug that plagues burst weapons in Fallout 2.
But to me, Fallout 1 simply does not have the feeling of great freedom and challenge of Fallout 2. You are rushed through it, and you have less freedom over your party. It is too damn short, and too damn easy. In Fallout 2, saving the world is a monumental task, and you do not simply breeze through it in a couple of real-world days.
In FO2 you have a huge world to explore. One of the things I like is how different sections of the map have very different types of encounters, with some areas simply too dangerous to set foot in with a young character. Fallout 1 had this with the Military Base area, but FO2 took the concept further. The overall difficulty level is much higher, and the proper challenge makes it more fun to me. You cannot just get a few good items and then walk through everything, as you pretty much can in FO1, but you really have to BUILD your character patiently to tackle the increasingly challenging tasks of the game. And you also have much more freedom about managing your party, which adds a fascinating element.
Some don't like the primitive tribals or the goofy stuff. I can see their point, but I'm not bothered by the goofy stuff. And I think it adds appeal to the game that you get to build your character from a primitive tribal into a modern one-man army. I like the idea that parts of the world really were struck back into Stone Age, and then you get to slowly work your way back...at the beginning, you have a spear and healing powder and you hunt geckos for food, rather than of being handed a 10mm Pistol and stimpacks right away like in FO1. In FO2, the 10mm is a valuable item that you have to wait some time for, and I like that.
In short: to me, Fallout 1 has always felt like just a well-made appetizer for the real adventure that is Fallout 2.
FO3 I have not tried yet, but plan to do so in the future.
I loved New Vegas, but after the first playthrough, the bugginess got really tiresome. Also, I do not appreciate the way these new Fallout games did combat. I used to be a hardcore 3d gamer, and the half-assed quake-wannabe 3d combat really got on my nerves. The FPS interface was not properly configurable, and it was inconsistent and buggy. Do it proper or don't do it at all, I say. I'd rather have a proper 3d gaming experience as in any modern FPS, or a purely brains-only combat system like the turn-based one of the original Fallouts. I also disliked the choices you were forced to make at the end of the game. Even so, I find the New Vegas is worthy of bearing the name of Fallout.