Read the script all the way through. Don't just read your character's lines. Read everything so that you know what is going on throughout the play.
Speak your lines out loud to get comfortable with how they sound. You can have someone read with you outside of rehearsal, or you can practice in front of a mirror.
Think about what each line means, and what emotions are behind it. Sometimes writers include notes to tell you the emotion of a particular line, but other times you have to figure this out based on context. The more you understand your character's background and motivations, the more convincing your acting will be.
Try to remember something in your own past that relates to the emotion in the character. If you need to be angry, for example, think of something that made you furious.
Think of your lines as other people deliver theirs on stage: do they seem like real reactions, or are you just reading lines? The more they sound like real reactions, the more in-character you will appear to the audience.
Practice until the lines seem to flow naturally, invested with the character's emotion and state of mind. Remember this frame of mind and use it to quickly get into character.