A big mistake made by inexperienced actors is memorizing lines with a specific rhythm and inflection. The more you say a line one specific way, the harder it becomes to do anything different with it. To illustrate this concept, try to conversationally speak the lyrics of your favorite song. You'll find you can't help but speak the words in the rhythm of the song. Because you've memorized the words in a certain pattern, your brain will be reluctant to break the habit.
Your only objective when learning lines is learning what to say and when to say them (your "cues"). How to say a line should come later, through script analysis, feedback from your director and interactions with your scene partners from one moment to the next. To being the process, make a concerted effort to learn your lines without any modulation.
A tried and true method for learning lines is having a friend, relative, or fellow actor "hold book" for you while you do a basic line-through. This method is good as it allows whomever you're working with to immediately correct any inaccuracies in your recitation. Make sure you're always saying the lines without inflection, and suggest your line-through partner do the same. The drawback to this method is that your memorization work will depend on another person's availability.
With technology, it's never been easier to record your own voice. Most computers have microphones built in, small tape and digital recorders are incredibly inexpensive, and many popular smartphones have voice memo features built in. Create a recording of yourself reading all the lines in a scene (or on a single page, for learning in more manageable chunks) except for your own. When you get to them, read your own lines under your breath at a normal conversational pace. This will create a pause in the recording after your cues long enough for you to fit your line into later. Listen to the recording as many times as you need to until you're able to pick up your clues quickly and accurately say your lines.