What makes a sequel good; not just enjoyable but also a proper, worthy and satisfying continuation? Seems to me that the key is partially in that last word, "continuation." There ought to be, first of all, a sense that the characters from the first story have grown, moved forward. Die Hard 2, for example, is great fun, but not much of a continuation. The Empire Strikes Back, on the other hand, absolutely continues the development of the main characters (okay, mainly Luke) and relies not only on the fact that you know who they had been, but required that knowledge to create the narrative.
At the same time, a sequel ought to offer an expanding and a deepening. You can expand the vision of the world the story takes place in (again, the Empire Strikes Back does this quite nicely as does, for instance, Beneath the Planet of the Apes), but more significantly, the themes that the first one explored should be expanded, given greater weight, examination and perspective. This expansion would, by its nature, also deepen one's understanding of the thematic undercurrents, which is crucial to a satisfying story. Are themes whose depths have already been plumbed in the first story really that resonant for being retread in the second? Notably, I don't think this thematic expansion necessarily requires that the story take on a larger scope. Indeed, it seems sometimes that if you make the sequel a tighter, more intense narrative it can aid in deeper explorations of character and theme. Philip Pullman's the Subtle Knife, sequel to the Golden Compass, is a great example of this.
In discussions of sequels, you often hear about how there is a demand to both give audiences/readers what they expect but also to surprise them. That has a very superficial ring to me. It seems to suggest that people want their expectations met but with a twist so they feel like they got their money's worth. Pushing this idea a little farther down the path, what you really have are audiences who want to visit characters they already know and like and be surprised by these characters in a way that feels fresh but also appropriate to their growth. Honestly, there's nothing about that that's particularly special to sequel-telling. Who doesn't want a story with good characters and a good surprise?