After receiving a new script and knowing what you have been cast as, the first thing actors generally do is look through their script and identify their lines. If you are worried about learning lines quickly, here are some tips that have helped me learn lines quickly.
Read the whole script, so you know the story and how your character fits into the plot.
Read the scenes you are in a few times. If you think you've done them enough, read it 3 more times. Read them out loud or in your head, whatever is most helpful.
Grab a friend who is going to be willing to help you. They will also have to be able to correct your words until they are perfect.
Practice in whatever accent you will be using for the show. Practice in your accent. Practice in your accent. You will be memorizing how you say things, and if you learn an entire script in your traditional way of saying things, and then you have to change to an accent you aren't familiar with, you will have to memorize the whole script again. Use your time well.
If you are playing a spouse, partner, or any part that has a main counterpart, ask them to run lines with you. This is a great way to get to know them better and form a friendship that can be translated easily to the stage. You can also start practicing how you may deliver a line.
All in all, memorizing things is personal and up to you. These methods are most helpful for me and I have been using them for as long as I have needed to memorize lines. Whatever way you learn lines, just make sure you are off book in time for when you are required to be so (which may be the first rehearsal, or the tenth). Be prepared to work on your feet and test your mind!
Break a leg with all your upcoming shows, auditions and rehearsals!
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