I often have to learn lines for plays and storytelling. There is no shortcut to learning words but these are some methods that I find useful.
1 Read the lines aloud. By speaking the lines you will hear them and they are more likely to stick.
2 Ask a friend to help you. Friends can correct you on any mistakes you make, give you the cue lines and go back over any weak areas.
3 Practise, practise, practise. This is the only way to make the lines stick. There is no such thing as a "photographic" memory. Everybody has to do this, even Kenneth Branagh.
4 Little and often. Go over them first thing in the morning, a few times during the day and last thing at night.
5 There are several apps which can help with learning lines. Here are some I have reviewed and recommend: With Line Learner you record all the lines including those of other characters and then listen to them leaving silent pauses to speak your own lines. With Rehearsal you can upload a script and watch it scrolling by as you record your lines to listen to.
6 Even if you don't use an app you can make a recording of the scene with a tape-recorder or mp3 player. Listen to it while you are shaving/washing up/driving (but keep your eyes on the road). It's a good idea to leave gaps in the recording to speak your own lines.
7 Move around while you are saying your lines. This has been scientifically proven to aid memory. The best thing to do is to act and feel the emotions of the character so that you are learning the meaning of the speech as much as the words. Or just for a change you can even do something entirely unrelated like juggling or sweeping the floor.
8 Go for a drive or better still a walk. Walking and saying your lines can be quite relaxing (though beware of strange looks from passers-by).
9 Learn the cue lines that lead in to each of your lines. Being prompt with your lines will give you and your fellow actors more confidence.
10 As you say or read the lines, follow the thought pattern of each speech and the overall progression of the scene. Your lines are a part of the play. They don't exist on their own.
11 In rehearsals, listen to and think about what the other actors are saying. Don't just concentrate on what you've got to say.
12 Make a recording of the cast reading the script and use this to practise with so that you get used to hearing the other characters' voices.