Practice, practice, practice. Writing out your lines is always helpful. For some, just the act of writing the lines down yourself aids in the memorizing process and places them into your subconscious memory.
When you receive your script, go through it, read it a few times and highlight your lines with yellow marker. Reading the script is important because you want to understand the story and how your character contributes to it.
Take the time to practice daily, at least 30 minutes to an hour each day. Don't just stand in one spot unless the script calls for it, move around as you speak, acting in character while you practice as though you are in the actual scene. Take the time to deliver your lines in different ways, project loudly, speak softly, use emotions, deliver the lines happy, deliver the lines sad. Even if you know your lines are to address sadness in the scene, delivering them as though you are excited or happy will not only create laughter and fun, it will help you to remember your lines as well.
If you have a lot of dialogue to remember, break your parts down in segments. Think about the scene and the character you are going to be presenting to, or how you will appear in the scene. Practicing parts of the script can be helpful.
Delivering scenes on a movie set is different than delivering a speech to a large audience. Often when delivering a speech you can read from notes or a tablet, but on a production set, you can't have notes in your hand. However in both instances you can occasionally get stage fright. It happens to the best of us, including me. If you make a mistake, laugh about it and keep going. We are all human and mistakes happen, so don't beat yourself up about it. More than anything, make the learning process fun by adding some silliness to the process.