Go scene by scene, writing your lines down in a notebook in all caps. Writing in caps is something we are not accustomed to, and so by concentrating on each letter we subconsciously memorize the words.
Pick out keywords. Memorizing every single word is not necessary if you are able to trigger the line after hearing a specific word.
Look for cues, especially when rehearsing. Perhaps someone turning their body around signals that your next line comes.
Stand in front of a mirror and watch yourself say the lines. For whatever reason, seeing your mouth move and utter the words helps in remembering what to say and when.
Understand the scenes. If you know that the specific scene involves a family argument, determine how you would feel in that situation. The script will usually reflect that in words.
Cover your lines with a paper and glance at the lines leading up to yours. After doing so, say your line that comes next. Remove the paper to check for accuracy.
Get a family member or friend to read with you. Psychologically, you will try to impress them with memorization skills and are therefore more likely to memorize.