Missed this bit on your post first time around so I'll answer now.
In the original, probably just the boat ticket quest. It's just a bit of a back and forth from Dobuita to the Hazuki Dojo. I see the forklifting job has been mentioned above, but I don't really agree. I found the forklifting pretty relaxing, but it's also broken up by the amount of fights and events the game throws at you during that part.
In Shenmue II, could be the books + chawan sign stuff? Maybe that's because there's only one way that quest resolves, even if there's actually a good handful of extra events that can be experienced in that time. I'm tempted to say Guilin too, but only when it hasn't been long since the last playthrough. Even then you can skip forward through a lot of it, and it's also the most soulful/mysterious/beautiful part of the game so...
In the case of either Shenmue game I stopped doing super quick playthroughs years ago. I like to take my time in both now, instead of going from main event to main event. I'm in no particular rush to get to any part of the game, so none of it particularly "drags" for me in the conventional sense.
