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I think the main question first is, what do you want to do with what you learn?
If you want to study a piece for the sake of absorbing material for your own compositions, you can approach it in a different way than if you are doing an essay/paper/etc on the piece for, say, music history analysis.
There's a lot of different approaches based on what goals you have.
If you want to take in the techniques used, then you should grab a text book that explains the theory behind the technique and then study literature examples (pieces) where the technique is used and then compare/check out how the technique is used across various different pieces.
If you're after historical/complete analysis, you have to dig up the history behind the piece (if available), the period it was written, composer and then go into other composers that existed at the time. Also important is that you look at what the "tendencies" were during that period, and how the piece you're looking at compares to those tendencies (Did it comply or was it a-typical? If so, why? Etc.)
Plus of course, an overview and analysis of the techniques (such as harmonic and rhythmic usage if applicable, or specific techniques such as 12 tone, serialism, clusters, extended techniques for instruments, etc.)
Then there's the analysis with focus on performance. This also depends if your goal is to do a historical reconstruction or if you're just figuring out how to give it your own spin and so on. A historical performance obviously requires you to look into historical sources (composer's own writings, letters, etc), books on the subject and so on to do something accurate within reason.
In all cases, it helps to have a firm grip of history and have more pieces around that you know from the same period. Analysis only gets "easier" the more pieces you analyze and the more music you actually learn. It helps enormously to have a frame of reference when faced with a new piece you want to analyze.
It's like, if you've already made your way through 20 or so baroque pieces, you'll be able to pick up a lot of trends and things by the 21th much faster than you were by the 1st. It's training your eye and sometimes ear to pick up specific things you know you have look out for depending on your objectives. Even if you're not shooting for anything specific, the nuances and details of things start to sink in the more you look and observe pieces and learn the historical/etc details by actually seeing it first hand in the music itself.
To further the example: If you're analyzing fugues and you've made your way through the baroque period, by the time you get to Hindemith or Satie's more out-there fugues you'll instantly pick up what they have and don't have from the tradition.
So really, think of what you want to accomplish and then arm yourself with what you need to do it. If it's a complete analysis, start with gathering historical sources, references, other pieces for comparison, and so on. If it's for picking up certain techniques, learning something specific, then figure out what is it that you want to look for and then gather pieces that demonstrate it in all possible ways, etc.
It's hard work, but it pays off in the long run.
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