Read the entire play first without worrying about memorizing your lines so that you understand it.
Get into the character, and feel the vibe of your role. Understand the meaning of the words and the motivations of your character in particular.
Concentrate on the context of your part in relation to the other actors, and the lines will flow naturally. Think about the lines in terms of communicating information to someone else to aid the memorization process.
Practice your lines while moving about instead of staying still. Researchers have found that actors retain their lines better when they add appropriate motions.
Memorize your lines one at a time. When you have mastered a line, repeat it and add the next line, and so on. The trick is to always repeat the previous lines and build from there.
Use word tricks when you encounter tricky spots in the script. Try rhyme, nonsense words, acronyms, gimmicks or visualization.
Cover up your lines with a paper, showing only the cue line before your dialogue. Read the cue, and try out your lines. If you are successful, then have a friend give you your cue lines.