Understand what expository dialogue is and how it works. The classic example goes like this: Guy A talks to Guy B at a pizza shop about his girlfriend. Guy A then works out his problems with his girlfriend.
He meets with Guy B later on, and begins their conversation with a line like "So, remember those problems with my girlfriend I told you about?"
Of course he remembers! It happened an hour ago, and the character is only bringing this up to advance the plot. This is sloppy writing and should be avoided at all costs.
An easy way to get around expository dialogue is to incorporate flashback sequences into the script. The television show "Oz" did this quite well. The flashbacks recap events so that the next scene can start sans exposition.
Flashbacks are not the most favored technique, however, so another way to avoid exposition is to write excellent action descriptions of the characters' body language and facial ticks.
Imagine, if after talking to his girlfriend, Guy A goes to meet with B once more, and Guy B has a look of anticipation on his face as if to say, "Well, how'd it go?" Right there, the audience knows what they are going to talk about.
Once the script is finished, give it to someone outside of the screenwriting business, explain exposition and ask them to outline areas that feature it rearing its ugly head.
Either eliminate the dialogue completely or make notes in the script that are merely placeholders for facial ticks and body language.