I generally approach a piece in three steps:
1. Examination of the score. This gives me a general idea of the challenges, the harmonic language, the structure, the dynamic peaks and valleys, etc. After I do this, if a recording is available, I might give it a listen, because only then can I really find details in the music like "oh, she's bringing out the tenor line here, or a little rubato here adds a hint of anticipation to what's about to happen, or a nonlegato touch really lightens things up here" and things like that. I do these things not to mimic them but to sort of get a deeper sense of the choices I can make, things to aspire to, things to disagree with, but mostly to refine my vision of the piece further.
2. Slow sight reading. This is really crucial for me to get a sense of how my fingers respond to the piece, to see what spots are tricky, and to try out some initial fingering.
3. Phrasal or sectional memorization. This really depends on the difficulty of the piece for me. If something is not so difficult, like a Mozart andante or one of Scriabin's easier preludes, I might learn by the page, by the section (if it's in something like ABA form), or until the repeat. If it's more difficult, I usually try to learn by the phrase, which is often about four measures. I like to do this because it's more musically complete, so it sticks in my memory better because I'm not stopping at some arbitrary point. For certain pieces this might not work as well (like if the piece is a moto perpetuo style stream of sixteenths). But even in these I like to learn in chunks of some sort. Sometimes these chunks are musical, like learning up to a crescendo or decrescendo, or learning up to a point where the figuration changes (up to where the figuration descends rather than ascends or something like that). But it can also be purely physical, a complete rotation, a moving in or out, a moment in something like an octave or double thirds passage where I can lift my hand a little and release tension, etc. My main goals when learning a piece are to learn it logically, whether it's in terms of the music and structure, whether it's about the figuration of the music, or whether it's something about my physical playing mechanism, because I feel if you're learning bar to bar or two bars at a time, it often imposes artificial divisions in the piece that aren't really there and can be counterproductive to playing the piece fluently (this said, there are situations in some music where a single bar or two might have a musical or physical significance).