For me, in chronological order, the best beginnings were:
Dark Sun: Shattered Lands. Gladiatorial beginnings are quite rare even these days, back then it was both rare and awesome. In the beginning they introduced you to setting via few memorable NPCs, you had quite a few C&C moments in the barracks and in choosing your escape, also I really liked the engine and level design. Wake of Ravager had more cold-open beginning, and while I like it more, bugs and all, DS:SL was better in the terms of beginning and overal story and C&C.
Daggerfall. I still remember that sample of closing and opening door, also ebony dagger and the imp. It wasn't very interesting in itself, but when you got out of the Privateer's Hold, you were struck by how enormous the world was, and how insignificant you were. I still remember that feeling.
Albion. Both the ship and the planet parts of the beginning were superb, they contrasted nicely with each other, highlighting the otherness of that planet. Graphics were nice for the time too.
Planescape: Torment. The best opening ever. The Mortuary itself, Morte, Dhall, Deionarra, Mark Morgan's music, MCA's writing, beautiful environments and design - it was just perfect.
Vampire the Masquerade: Bloodlines. The only criticism I have it's that the could've made different short Embrace stories for each clan, I especially laughed when your Sire, being a Nosferatu, seduced you. But other than that - it was perfect. Both the trial, the Sabbat incursion, Smiling Jack and all. And Santa-Monica. Ahhh.
Witcher 1. I was a fan of Ведьмак novels long before CDPR managed to make a game, the series were hugely popular in Russia among fantasy fans. The most pleasant surorise for me was that they managed t stay true to most characters, other than Triss, for whom a plunging neckline was uncharacteristic (and who was kinda badly voiced in RU version). But other than that, Kaer Morhen was great, and I liked the outskirts of Wyzima. Witcher 2 also had a good beginning, but lacked in novelty. Still good though.
While I love Fallouts, neither the first one's, nor the second's beginning have left an "AWESOME" impression. Great games, decent beginnings, but not great ones. The beginnings of both BG1 and BG2 were serviceable, the second one elevated by the performance of Irenicus' VA. The only interesting idea Bioware had for the beginnings was DA:O, and look how that turned out.