It really depends on the style of the music you're looking for so this is what you look for first.
1. You need to realise what the music you're looking at is "made off", or what era it says it is. Beethoven writes differently than Ligeti, or Bach. A twelve tone piece will be differently approached than a tonal one, or something else, etc.
So this is your first big step. This usually does take a bit of time, and also includes further steps as well (I mean, if you've seen that it's a tonal piece, you'll probably also have noticed the tonality, and the changes elsewhere).
2. In tonal music, you should go for a harmonic analysis. I IV I 6/4 V and I etc... If something goes to a weird place... well it means that a modulation has taken place, plainly. Where and how it is up to you to find out. Just keep in mind that for a modulation to actually "work" successfully, a cadenza is also needed, no matter how small. You can't establish tonality without saying V I in some way.
3. You will notice repeated elements here and there. You can very well spot some phrases which are repeating themselves here and there, or maybe changing themes (1st and 2nd in a sonata, for example). After that larger and larger, to which at a point you'll realise the full form of the piece.
The 3rd step could also be 2nd very well. It's up to you. In short it's what people call "FORM". A sonata? A fugue? A scherzo? What is this? It's most important, because you can expect things afterwards and actualyl skip large chunks of the piece!
I mean recurring themes don't take too much time, but if you don't know why they are there, or if they are in fact recurring then...
In non tonal music, the 1st step is rather important. You will need to realise what's going on and if there is something that "sticks out" sort of saying. Coherence is most important in composition and so things that are repeated (could be a pattern, could be an interval, could be a series, could be a rhythmic part, could be an orchestration quirk, whatever really), are the ones that will define what you will do next..