Good:
Morrowind. You're in the first village, do a quest or two, have a dungeon right in front of the village, can go for a swim and discover more dungeons on the islands, and then you're essentially free to go wherever you want. Go to Balmora by foot or take the silt strider. Tutorial is just a bunch of pop up windows telling you the controls, and then you're free to go. The time between character creation and being free to go out into the world to do your thing is less than a minute. Fucking great.
New Vegas. Same here, except you can do a little more stuff in the starting village. Then you can go and explore wherever you want. Actual tutorial time is about a minute, too.
BG2. I actually like Irenicus' dungeon. Yeah, it can get repetitive after the 5th time or what, but it's a really cool and interesting place to explore the first time around and you might even miss some stuff so the second time around will be fun too. It's deeply connected to the main story and even has some stuff for side quests in it (like the acorns you can get from the dryads if you help them), it's a high quality dungeon on par with the game's other dungeons.
Arcanum. Zeppelin crash, some weird religious guy tells you you're some legendary figure's reincarnation, you're confused about wtf is going on - then you can explore the crash site, find your first quest which already has a little choice to make and shows you that NPCs might lie to you, or you can leave straight away for Shrouded Hills and do the quests there - which is a great starting town with a handful of quests to do, choices to make, characters to meet. The perfect starting place before you're going to the massive capital city of Tarant.
Fallout 1. Some easy rats to beat up, then you leave straight away for the first town and are on your own. The rats take you maybe 2 minutes, then the good part of the game starts straight away. Nice.
Age of Decadence. Start in a small town where you choose your faction and do some sidequests. There are many issues I have with AoD's structure and forcing you into one specific path that often pulls you into the next step of its main quest without enough warning, but the beginning area is a nice demonstration of what the game is gonna be like and already gives you some major choices to make.
VtM: Bloodlines. You can either go through a small tutorial with Jack, who's a cool character, or... you can skip the tutorial entirely! Nice! It's a decent tutorial but if you don't wanna do a tutorial at all you can just skip it, perfect for replays.
Gothic 2. Short conversation with Xardas, then you're off to explore and find a way into the city. You're weak as fuck and will die easily if more than one wolf gangs up on you. You will die easily if you try to defeat the bandits that prey on you right on the first fucking road. None of the NPCs care about you, you're not going to get in the city unless you do some work or fork over some cash - or are good at exploring and find an alternate way in. Perfect.
Bad:
Fallout 2. Temple of Trials can be gotten over with quickly, but it's not really interesting or engaging. Literally just a tutorial area without anything interesting to offer, unlike Irenicus' dungeon.
Skyrim. It just takes so much time to sit through. Bethesda's design of openings has declined massively since Morrowind. Daggerfall had you start in a cool little dungeon straight away, Morrowind had you start in a small village straight away, Skyrim has you sit through 10 minutes of unskippable in-game cutscene before you can get started. Yawn. I wanna play a game, not watch a movie, what is this faggotry.
NWN2. Boring village, boring characters, essentially just being confronted with 500 generic fantasy cliches before you get to do anything. Yaaaawn.
Pillars of Eternity. It's a pretty banal beginning. Hey you got the shits or something, go find some herbs to stop your diarrhea. Nothing of consequence happens in this beginning scene, it's just wasting your time on trivial things. Meh.
ToEE. I love the game and its combat, but the problem with Hommlett is that it offers you very little of said combat, just some boring fetch quests of no real consequence and with no real choices attached to them. Very weak display from Troika after Arcanum's great opening area. Good thing the Co8 mod offers you a nice alternative to the fetch quests in Hommlett by offering you a small dungeon crawl quest to level you up.