Note: I started this as a reply to a comment on my review of Marooned in Realtime, but it grew to the point where I decided to split it out into its own post.
Well, the most important Asimov to read is the Robot-Empire-Foundation series (which is actually three separate series that he joined into the same universe after the fact). Wiki can give you the full list in chronological order, but I think how you read them depends on what you’re looking for.
In my opinion, the natural entry points are Caves of Steel (the first robot mystery), Foundation (the first foundation book published), or Prelude to Foundation (the first foundation book chronologically, at least as far as the ones written by Asimov himself go). I didn’t include The Currents of Space, the first Empire book, because I really think the Empire books are the weakest of the three series and I mostly read them as background for the Foundation books that came after the series were connected.
Caves is a detective novel in a sci-fi setting, more or less. It posits an interesting future Earth where the cities have been domed over and the land in between is reserved for robot agriculture. At this point Earth has also colonized many exoplanets and the people on them have created a distinct culture that is very different from Earth’s. Most of the conflict in the book is between Earth and Spacer culture and centers around the use of robots, which Spacers take for granted but Earthers only accept in a very racist/slave owner sort of way.
Foundation is probably the most famous book Asimov ever wrote, even though it’s actually a collection of serial shorts he wrote over the course of a decade. In some ways, it is like Marooned because it tries to capture human events on an epic time scale. The difference is that Foundation is concerned with the course of civilizations rather than individuals. The galactic empire is slowly falling, and Foundation is mostly about how you pick up the pieces of humanity after decline and fall.
Prelude to Foundation is exactly what it sounds like, a novel dedicated to the events before Foundation itself. It tells the story of a young Hari Seldon, the genius who tried to mitigate the damage of the collapsing empire. The reason I’d include it as one of the entry points to the series isn’t as much about chronology as it is about the story itself. Asimov’s early work was pretty rough by modern standards. It was all about the ideas, and things like characterization were not handled all that well. In addition, because Asimov’s ideas were so wonderful they have been copied and expanded upon since his original versions came out. They tend to lose some power now because you think “well, that’s just like x” except that you have to stop and realize that Asimov was the one who did it first and made x possible. Prelude, however, was written in the 80’s when Asimov was a much more complete storyteller, so it is more accessible to modern readers than Foundation might be.
Other starter Asimov that’s worth looking at are the robot short stories (collected in I, Robot among many others, which has absolutely nothing whatsoever to do with the movie), The Gods Themselves, Nightfall, or Azazel (for something a little different). Personally, I think a lot of his best work is in his short stories, so grabbing a collection is not a bad way to get into his writing.