As Dr. Thomas Burritt likes to put it, "Where does this music live?" In the huge world of music, where does this piece fit? (Hear more about this from Dr. Burritt's discussion with the @ percussion podcast crew -- starting at 22:00.)
What style is this music? What are characteristics of this style?
What are other pieces that live in this same realm or style? Can those pieces inform how I approach this one?
Who are some experts in this style or genre of music? Have I listened to them perform?
What does the title of the piece communicate?
What does the tempo marking (if there is one) communicate (Tony Cirone won't let you get away with not knowing!)? (Also see here for more.)
What is an appropriate tempo for performance? For practicing? How do I choose?
Is this piece a transcription or arrangement from another instrument? If so, what do I need to understand about the original instrument for which the music was written?
Who is the composer? What do I need to know about the composer? What else has s/he written? What is his/her background? Where is s/he from? Why does this matter?
If I chose this piece of music myself, why did I choose it?
If my teacher chose this piece for me to play, why did s/he choose it?
Are there other musicians available to me who can provide valuable insight into the music I am learning?
If this style or genre of music is unfamiliar to me, what elements of it can I relate to or do I know something about?
If I'm performing this as part of a recital or concert of other music, how does this fit in or relate to the other pieces?
If it's an ensemble work, how does my part fit into the other parts? What is my role in the music? What else is happening at any given point in the piece?
What do I hope to communicate through this piece? An emotion? A story? Novelty? A scene?
What does the composer intend to communicate through this piece? Are there program/performance notes available?
Is it my top priority to focus on the composer's intent or on my intent (if they are not 100% in line with each other)?
Who is my audience? Does this affect how I perform the piece?
What is the venue in which I will perform this music? Does that affect my choice of implements? Tempo? Phrasing and expression? (For example, a large reverberant hall may call for a slightly slower tempo or harder sticks than the same piece in a small, dry venue.)
What choice of instrument(s) and implement(s) best suits the nature of this piece and the performance I plan to give? How do these choices affect the sound and ultimately what the audience experiences?
How is the piece organized? What is the form of the piece? Is there a clear form?
What is/are the hardest part(s) of the piece? Easiest part(s) of the piece? Does this affect the order in which I choose to learn the music?
What patterns are present in the music?
What are the similarities and differences among different sections of the piece? Does this matter?
What technical challenges does this music present that I need to develop?
What technical exercises can I practice to develop the necessary technique for this piece?
How does timbre affect what I'm trying to communicate? How can I adjust the timbre if necessary?
Do I need to consider adjusting my grip, motion, posture, or other physical element in order to best communicate the sound I want?
Are there multiple voices sounding simultaneously? If so, how do I balance these? Is one most important?
Are certain elements of the piece (or passage) more important than others? Why? How do I ensure those elements are communicated as most important through my performance?
How important is steady tempo? Do I have freedom to adjust tempo? If so, what determines how I adjust tempo?
What scales/keys/harmonies/pitch sets are present? What can I do to ensure my comfort with those?
What rhythmic elements and meters are present? How can I practice the new/challenging ones?
What am I doing to make this performance unique to me and not a copy of someone else's performance?
Am I comparing my interpretation of this piece to someone else? If so, who? Why?
(After recording yourself and listening/watching back) What is working and what is not working in my performance and communication of the piece? Why?
What do my motions/gestures communicate musically? Does my movement match the expression of music I intend or are they conflicting? (More on gesture here.)
As an experiment, what are three (or more) different ways I can play a given passage? Which one best communicates what I want to say?
What am I doing to communicate cohesive phrases of music? (Sentences/paragraphs/story as opposed to letters and words.)
I could probably keep going, but this list will keep you or your students busy for awhile! Are there other important questions to ask about the music that I am missing?