The first step in the process is to take the script and mark your lines with a colored highlighter. This will serve as a visual cue that will make it easier to remember where your lines occur in the script.
Read and reread your lines, as frequently as possible. In addition to learning your own lines, it's also important to learn those of the other actors in relation to your own lines so you'll be aware of the cues for your character's dialogue.
Just as a script is divided up into acts and scenes, an actor can further divide up the text into smaller chunks that can be more easy to memorize than attempting to memorize the entire script at once. When rehearsing for a play, directors often work on specific scenes before attempting a run-through of the entire play. Memorizing lines for each scene as it will be rehearsed can be an effective method of eventually memorizing the entire script over time.
Reading a script the way you would read a book is not necessarily the most effective way to memorize lines. Since you'll be speaking the lines out loud, it makes sense to rehearse out loud as well. Many actors find it useful to rehearse their lines with a friend or family member. Having another person run lines with you in context can help provide the auditory cues, with the end of her lines reminding you of the first words of your own lines.
Another helpful memorization technique is to use a cassette recorder or digital recorder to record yourself reciting your lines. Then, listen back to the recording repeatedly while doing other things, such as driving or household chores. This can complement the other memorization efforts you undertake; and can also help you learn the lines subliminally by letting them seep into your subconscious when you're not specifically working on memorizing the script.