Usually I listen to the piece a few times in recordings by different people until I can hear portions of it without the recording on, and have a general idea of the structure. Exceptions are music with no recordings or only one recording available (new music!) and pieces I learn with my current teacher who is very much against listening before learning the piece.
After that, if the piece is easier than a Beethoven sonata I start by sight-reading the whole thing then breaking it down: picking a page at a time and working on melody lines alone, inner voices alone, getting into detail with fingerings and articulations, planning phrasing.
If the piece is harder than a Beethoven sonata I go much more slowly, sticking only to the first section until all notes are learned and fingering is functional, still with a lot of individual voices alone, then moving on to the next section. Learning speed depends how much time I have. The most difficult piece I learned this year was probably about a page every week.
For fast pieces I wait until very late in the learning process to start increasing the tempo. Except once in a while trying performance tempo for fun