I've found that, when it comes to learning new music, there are two types of people out there: those who get a complete thrill from exploring and learning new things, and people who find the music-learning process about as exciting as going to the dentist.
Sometimes, there are circumstances beyond our control that make the music-learning process difficult, but we can often mitigate these with a good music-learning system!
Below, I outline a basic music-learning system that can be employed for most musical theater pieces, as well as pieces in other genres. This isn't the *only* way to learn new music, but it is a way that I've found helps you to internalize and connect with the piece very quickly.
STEP ONE: GET THE SHEET MUSIC
This might seem painfully obvious, but you MUST get sheet music for your all of your songs. While sheet music, with its many lines and symbols and strange Italian words, might look intimidating to a beginning student, it is essential because it is literally the entire blueprint for the piece. It contains about 90% of the information you need to learn, interpret, and perform the piece.
Your sheet music should be a piano/vocal score in the key you plan to sing. A simple lyric sheet, even with chords, will not suffice. If you're not sure how to determine what key is best for you, ask your teacher or coach.
STEP TWO: RESEARCH THE SHOW
Before you begin to learn the piece, you need to be familiar with the show your song is from and the character who sings it. You will not be able to get far in your interpretive process if you don't know the context of your song. mtishows.com has great synopses for tons of shows; there is also good ol' Wikipedia.
WHAT *NOT* TO DO NEXT
After you've completed steps one and two, your first instinct might be to listen to a recording. Unless you are in a position where you need to learn the piece VERY quickly, I would advise strongly against this. There's a good chance you'll simply end up imitating what you hear, possibly without even realizing it. While it's okay (and sometimes necessary) to use recordings for reference or comparison, they should NEVER be your primary practice tool.
Or maybe, if you have some keyboard skills, your first instinct might be to sit down at the piano and teach yourself the notes. While this is certainly a necessary step and your instinct is to be commended, you should not do this just yet. To understand why, I encourage you to think about how musical theater songs are created. Which brings me to step three.
STEP 3: LOOK AT THE TEXT
Musical theater composers almost always write a song based on an existing text. The lyrics are written first, and then the music. It is virtually never the other way around.
Since the text was likely the starting point for the composer, it should be your starting point, too. Treat the text as you would a monologue. Figure out where the dramatic beats are. Decide what the character's intention is for each beat. Play around with different inflections and emphases. Do everything you can to make this text YOURS. That way, when you finally begin learning the music, you will have a very clear idea of the dramatic choices you want to make. (Bonus: the text will already be memorized, or at least very close to it.)
Even if you're already somewhat - or even very - familiar with the piece, I'd still encourage you to complete these first three steps before singing it. Gaining basic familiarity with a piece from hearing it in performance or on a recording is one thing; it's another to take it apart yourself and make your own discoveries and decisions about it.
STEP FOUR: LEARN THE MUSIC
After you've internalized the text, THEN you may begin learning the music. Start by learning your vocal line. Preferably, you should do this yourself at the piano. If your keyboard skills aren't too strong, ask your teacher or your coach for help. Make sure you also know the meaning of all musical terms and symbols in the score.
Listening to a recording may be helpful at this stage, but make sure you choose a good one. Your teacher or coach may have suggestions. While you're listening, be sure to follow along with your own score.
STEP FIVE: BEGIN INTERPRETING
Interpretation is a multi-faceted process to which I could devote another entire post. For now, though:
As you're practicing and working your vocal technique into the piece, start to think about how
you want to perform it: your gestures, your facial expressions, your visual focal points. Your goal is to make clear choices that convey the drama. When you begin this process, start by considering what you know about the character and context of the piece, then look at the piece itself. Consider the mood(s) of the song, its overall structure, and the dramatic arc (i.e., the character's journey in this particular piece).
A couple important things to keep in mind about interpreting:
Often, this process is never really "done." We may make certain choices at first, then return to the piece months or years later and discover that our opinions have changed. This is completely legitimate, as long as you can justify your new ideas.
Sometimes, there is no one right answer. This doesn't mean that anything goes (again, you must have good reasons for your choices), but it does mean that this process is somewhat subjective. Your opinion might be slightly, or very, different from someone else's, and that is okay.
Just remember, no system is absolutely perfect for every piece - some songs will require different processes - but these basic elements should always be part of your process in some way. So give this approach a try! I suspect that you will find greater connection to the piece because of it, and that you will memorize it faster (who wouldn't want that?!).