The way I go about learning a new piece is similar to the way actors prepare to play a part in a tragedy, because after all music is another language; one tangled with deep emotion.  For a performance to be memorable the emotion and expression has to come through clearly.  This applies across the board, whether it is a piece from Suzuki Method: Book 2 up until you are playing like Hilary Hahn.  Also we should look to science to see if it can help us with the fundamentals in learning music.  You MUST restrain yourself from playing a song right off the get go because there are critical steps that must come first if you want to learn the piece quickly and be able to shape the dynamics of the music to your liking. 

I recommend the following steps for each and every performance piece you are learning:

1.  Listen, Listen, Listen, and listen and listen.  And then listen more!  To every recording (especially performances, youtube is great for this!) you can get your hands on and pay attention to two things: the basic melody and rhythm of the music so when you look at the piece you already know what it sounds like and also look/listen to the mood that the performer is affecting by his/her expression (you will be more concerned about this a little later once you have learned the piece).

2.  Once you have listened to the song enough you will have the song "in your ear" and this is exactly what you need before you even look at the sheet music (whenever possible).  Compare this to trying to read Shakespeare by yourself: ("sight reading") reading it properly the first time you see it or watching his play live hundreds of times (watching and listening to others perform). *Note here I am not bashing sight reading at all although I think that it has its time and place and if you truly want to learn a song quickly and efficiently it is a dangerous way to proceed (especially if you are not that good at sight reading yet!)

3.  Okay now before we play there is something that is very important to do to the sheet music.  Either ask someone else or try to guess at it yourself at where the beginning, middle and end of the song (also called introduction, development, and recapitulation) are and make a little note of this (if you listen to the song enough it usually isn't that hard).  This is important for you mentally and emotionally as traditionally each section involves a distinct "sound/mood" and tells a different part of the story and it is incredibly important for you as the performer to have these mental ques developed from the start.

4. Next, listening to your favorite recording of the song and holding the sheet music, divide the music into phrases (in our language equivalent these are like sentences) because this is how you will finally learn the song. (***Notice how we didn't even play a note yet!)  This process could take a couple weeks if done properly and you are learning this skill.

5.  Lastly, before you attempt to play the song you should look over the song (and you will get a fairly good idea from watching others perform) to see if there are any techniques that you feel you are not proficient at or do not even have the ability yet to perform.  If you need to work on the technique I suggest finding an etude for it or my favorite, incorporating it into the scale of key that your song is in (this way you work on intonation and technique all in one).  Trust me it is far better to ensure that your technique is there before you begin learning a piece because there is nothing more frustrating than getting half way through a song (many times the most technically challenging, ex: Czardas with artificial harmonics in the middle).

6.  Okay, once steps 1-5 have been executed (and WELL!) you can proceed to learning the piece in the phrases that you had made before.  One really important thing here is that THERE IS NO RULE ON LEARNING A SONG FROM START TO FINISH!  I cannot stress this enough.  Learn the phrases which come to you and "feel" the best when you play them and only tackle the harder ones when you are feeling very confident and energetic because if you force yourself to learn them before your ear/technique are ready you will develop bad habits which WILL be excruciating to break.

7.  Once you have learned the song to a level where you can play the phrases technically well you start to play around with emphasis of different notes and "heaviness" of the bowing to add character (Fiddlerman had a video on this).  It is very important at this stage to record yourself and listen back as well as start performing it for close friends as they will have a fresh perspective (because they haven't listened to it 3000 times ).  Remember, music is about the story you are telling to the audience; you are playing to the audience not "at" them.

--> So just to end off, I hope I don't bore anyone these are just some lessons I have learned the painfully long way and would not want want anyone else to have to endure.  The also revolve around the science of how we learn to do things.  Scientists claim that our motor engrams (patterns associated with muscle movements) take 300-500 repetitions to learn but get this... it takes 3000-5000 reps to break an already formed engram.  What this means for practicing is that every time you play you play perfectly and who knows you literally could learn an entire new piece, polished in a month!  Remember: "Practice does not make perfect, perfect practice makes perfect"