Most actors will underline or highlight the lines for their role before the first read-through with the director and cast. To help you get a jump start on memorizing lines, before you meet with the group, read the play at your leisure at least three times, from front to back. Doing so gives you the total overview of the play. By the time you attend the first rehearsal, you will already have begun to learn lines without even trying.
Knowing the entire play will also help to 'cover' or 'pick up' when other actors 'drop' lines on stage. So, study the script. Read your lines over and over again, as part of the entire scene so you get them in context. If you find you always tend to forget a particular line, mark it or highlight it and use a sticky-note to return to later.
Tape record a rehearsal or two and play back parts to practice lines on your own time. During breaks at rehearsals, practice lines with other actors backstage, and record the scene you work on. Listen carefully to playbacks. It's all about connecting with the meaning of what you are saying, as well as just recitations. If you truly comprehend your character, the lines will become second nature.
A great way to learn lines is to write them all out. Use index cards or blank lined paper. You may learn the entire play as you will become more clear about other character's cues to your own lines.
Breaking the lines down to comprehend each scene will help you absorb the meaning of the lines you are expected to deliver.
Practice lines before the mirror. Talk to your reflection in character. Your face muscles will actually begin to 'remember' the speech patterns you use, and whatever affectations you apply. Repeat the same gestures with the same lines to help key you to any you stumbling blocks. Sometimes just doing the gesture will spark the line back into your memory. You can also tape your index cards around the mirror edges--try it with the cards facing away from you and flip them over only if you need to peek.
Sing the lines. For a funny scene, make a funny or even ridiculous tune, using the lines. For more drama, try heavier, deeper notes. For a hard to remember monologue, find the rhythm of the speech, then repeat it in song. You will quickly realize if something is missing when the rhythm seems off.
Run lines with other actors when not at specific rehearsal periods. Choose one scene. Really listen to what is being said, rather than just wait for your turn. Hold the play and read it together several times. Then go 'off book' and say the actual lines. Try doing the scene without using any of the playwright's words; tell the story in your own words to be sure you are clear on what is transpiring. Then return to the original script, and re-run the actual lines.
When on stage, trust that you will remember everything. Always act confident, and if on the off chance you should ever forget a line, just pause a second, take a deep breath and think about what is going on. The drama of a suspended moment is often enough to have the audience in the palm of your hand.