The Star Wars Expanded Universe can seem impossibly huge to fans just starting to explore it. As of 2010, the EU spans over 5,000 years and contains over 1,500 novels, short stories, comics, video games, TV episodes, and other types of media.
Going through the entire Expanded Universe according to in-universe chronological order (that is, starting with stories covering the earliest history of the Star Wars galaxy) would be difficult because it is so extensive and continuously expanding -- and because some of the earlier EU material is difficult to find. There are a few different ways to approach the Expanded Universe, however, to make it seem less daunting.
1. Read in Order of Publication
If you're concerned about missing important events or characters, reading the Expanded Universe in order of publication is better than trying to read it according to in-universe chronological order. Things may jump around a lot in the timeline, but you won't accidentally read spoilers or miss references to previous events -- since authors can't reference things that haven't happened yet.
The tricky aspect to reading in publication order, however, is deciding where to start. Much of the early Expanded Universe -- for example, Splinter of the Mind's Eye and Marvel Star Wars -- has a different feel than the later EU and is not entirely canonical. Other parts, such as the Ewoks and Droids cartoons, may be difficult to find. Your best bet is probably to start with Heir to the Empire by Timothy Zahn (the first novel in what's called The Thrawn Trilogy), which kicked off the Expanded Universe as we know it today, and return to the earlier material later.
2. Read about Characters You Like
Since the Star Wars universe spans so much time, it can't all focus on the same characters. While the main focus of the EU novels has been the Star Wars movie characters, even the early Expanded Universe explored side characters in more detail: for example, in the Ewok Adventure movies, which star Wicket the Ewok, or Tales from the Mos Eisley Cantina, which features background characters from the cantina scene in A New Hope.
If you like one character or group of characters, that can help narrow down the Expanded Universe for you. Try looking up all of the character's appearances, then reading in chronological order.
3. Read Your Favorite Format
The Star Wars Expanded Universe contains stories in a variety of different formats. (Indeed, using "read" as shorthand is not entirely correct, since you can't read a TV show, video game or radio drama.) Each format has its own trends, with some exceptions: for example, Star Wars comics tend to star original characters and explore separate storylines and parts of the timeline than Star Wars novels.
Paying attention to only one format in the Expanded Universe may mean you miss things; for example, the comic Dark Empire sets up some important events and character development for the Jedi Academy trilogy of novels. Even when this happens, though, there's usually enough introductory material or narration to explain what you missed.
4. Read What You Can Find
In the end, reading the EU in a particular order may not be worth worrying about. Most of the stories and series are structured so that you can read and understand them even without too much knowledge of the rest of the Star Wars universe, and are perfectly enjoyable just as stand-alone tie-ins to the Star Wars movies.
Even if you don't fully understand something in the Expanded Universe, there are plenty of resources to help you get background information, such as the Star Wars Databank and Wookieepedia. Start with a fan favorite, such as The Thrawn Trilogy or the Revenge of the Sith novelization, or just jump right in with whatever Star Wars media you can find in your local library. There's nothing to be afraid of!