If you liked the original Star Wars series and want more of it, the novelizations are perfect for you. While the basic narrative of the films remains the same, there is more going on in these books than you see in the films. Characters and their back stories are fleshed out. There are more scenes in the novelizations of the films than there are in the films themselves. You can also read the books and look for minor differences between the books and the original films.
Splinter of the Mind's Eye is the first Star Wars tie-in novel, written all the way back in 1978. The novel was written by Alan Dean Foster, who ghostwrote the novelization of the original Star Wars film. "Splinter" is interesting because it was written before the mythology of the Star Wars Universe had fully been fleshed out. Thus, you'll see some interactions between Luke and Leia that are a little strange considering what we find out about them in Return of the Jedi. While there isn't much in this book that ties in to the rest of the series, it will be interesting for Star Wars fans as a historical curiosity and anachronism.
The Han Solo Adventures is not to be confused with The Han Solo Trilogy. The former was written in the late-70s /early-80s and tells the story of Han Solo as an adult smuggler, while the latter tells the story of his early adulthood and childhood. The Han Solo Adventures is a great piece of writing from Brian Daley from when there weren't a lot of licensed Star Wars novels coming down the pike. Those interested in Han's years as a rogue and a smuggler will be particularly drawn to this series. It is available in a collected paperback edition, which includes all three novels.
The Thrawn Trilogy is noteworthy for being one of the first Star Wars adaptations released many years after the original films. Timothy Zahn doesn't depart too much from the original trilogy. He tells the story of the enterprising Grand Admiral Thrawn, who rallies the remnants of the Empire and mounts a vicious attack on the newborn Republic (which is run by many of the heroes of the Star Wars films). There is plenty here for fans of the movies, including the return of the Empire and copious amounts of Force-related mysticism.