Okay so Asimov wrote three separate series that all take place within the same universe. The best way to read them is in chronological order of actual events within the story. The three different series inlcude the "Robot" series, the "Emipire" series and the "Foundation" series. The last book of the "Foundation" series wraps up all the loose ends of the series and there are numerous other refreneces that justify reading the books in this order.
The "Robot" series starts off with an excellent book of short stories. Asimov was just as talented at writing short stories as full length novels and I, Robot inlcudes some of the best. I, Robot The first story takes place in 1998 and introduces us to the "three laws of robotics" which run throughout all the books. The rest of the stories take place approximately over the next 80 years. The crux of the "Robot" series continues over the next three books, The Caves of Steel The Caves of Steel (R. Daneel Olivaw, Book 1), The Naked Sun The Naked Sun and The Robots of Dawn The Robots of Dawn. These are detective novels at heart, following the adventures of a human detective Elijah Bailey and his robot partner Daneel Olivaw. The series finishes with Robots and Empire Robots and Empirewhich concludes some of the major questions and leaves some things unexplained.
Asimov's universe picks up again later in the "Empire" series. Asimov considered this to be some of his weakest writing and if you were going to skip one series this is the only one to consider. It's not that these books are bad, its just that they are not up to the normal level of Asimov excellence. They are still out of print but floating around the internet and used bookstores. The Stars Like Dust The Stars, Like Dust (The Empire Novels), The Currents of Space The Currents of Space (The Empire Novels), and Pebble In the Sky Pebble in the Sky (The Empire Novels).
Asimov's universe concludes with the fantastic "Foundation" series voted Best Sci-Fi/Fantasy series of all time. "Foundation" takes places thousands of years in the future and deals with a genius using "psychohistory" to predict and save the future of mankind. It also deals with various other themes and universe specific issues and Asimov's writing strength is his ability to use simplicity and conversation to pace the events and action. The first book is Foundation Foundation (Foundation Novels) followed by Foundation and Empire Foundation and Empire (Foundation Novels), Second Foundation Second Foundation (Foundation Novels), Foundation's Edge Foundation's Edge (Foundation Novels) and Foundation and Earth Foundation and Earth.
The books in this series have been nominated and won countless Hugo awards for best sci-fi/fantasy books and won the Hugo for Best Series of All Time. One of the real strengths of the series is the way Asimov infuses detective/mystery and thriller genres into his science-fiction world. It works very well and provides variety to keep the large series interesting. Like all good books, Asimov writes brilliantly thought out stories, but also addresses many larger societal issues that are thought provoking and will keep this series important for years to come.