Sequels are sometimes an excellent way to subvert the expectations of an audience. I really like those types of sequels. People mentioned Monkey Island 2, and I can imagine somebody doesn't like it because of the ending, but that's why I like it. It's a very challenging, idea, isn't it? It's up to you to decide what it means, but it ends up bringing about an entirely different context for the original depending on how you interpret. Other left-field sequels I appreciate include Silent Hill 2, Metal Gear Solid 2 (especially this one! What a subversive game!) and Ultima IV, which of you think about it was a very disruptive sequel that launched the series down a new and ultimately more satisfying direction. Sometimes you get cases like Resident Evil 4 were people love the sequel, but don't necessarily like where it's sequels went with the games.
It is, however, a pity when you get games like Ultima IX or King's Quest VIII, because it's such a waste of time of perhaps talented programmers and designers who could have been devoting their years of work to something better. That's always a shame.
As for sequels being better than the originals, it happens all the time. The Hand of Fate is so much better than any other Kyrandia title, it's ridiculous. Might & Magic II is such a humongous improvement on Might & Magic, the only possible debate for me is that the sequel is maybe too crazy and the original is an originator, so there's always some charm in that. I think everyone is happy that Devil May Cry 3 happened and didn't leave the series withered away at 2. It's not a very popular opinion, I know, but if you ask me Baldur's Gate is an exceedingly mediocre RPG that only really found popularity because it had a well thought-out, progressive engine, but the design was really grating and flabby in a lot of places. I've not met many people who like the original better than the sequel.
Reboots and remakes, on the other hand, would be made much better if there wasn't the issue of copyright. If properties' IP rights died sooner and people could reinterpret the stories in the their own way, like people making different versions of Hamlet or Robin Hood, putting them into the discourse of public legend, I'd be much more enthusiastic. As it sounds now, the issue of money and ownership makes everything such a walled garden that these are often not very promising endeavors. Just imagining what would have happened if fairy tales had been birthed in the days of intellectual property rights makes me frowny.
When it comes to games though because there is the issue of technology, remakes tend to be a fun way to get a "what-if" scenario about the game. I prefer two things: 1) a design team close in make-up to the original to handle it and 2) them to go hog wild with ideas for how they want the remake to be. The original will ALWAYS be there. If it's famous enough, there's very little chance it will fade into history, with the remake being the only thing that survives. Therefore I prefer it when creators do something different with remakes.
I know this is primarily a PC gaming forum and I did enjoy Sierra's remakes, as well as the AGD versions (especially the King's Quest II remake, that was incredible), but my favorites are things like Wild Arms: Alter Code F or New Etrian Odyssey (Shin Sekaiju no Meikyuu) where the remakes are these complete re-envisioning based on the original ideas. I prefer, in this case, for the original to be included with the remake, but it's not a deal breaker for me.