I'm "untrained" aside from the 5 years of instruction I had while a student in college. However I don't feel like I was given the necessary tools at that time - just slammed with a bunch of music and told "you have a recital at the end of the semester, you need to learn this".
My approach is this:
1. Play through (sight read) the piece to see what it entails.
2. If it is playable, or could be playable at my current level, I start going through the piece, playing it repeatedly.
3. Part of that is to become familiar with the piece (assuming I haven't heard it) and do things like identify any tricky spots, any surprise fingerings or fret positions, and just get the general shape and feel of what's going on.
at this point I may:
A. Just simply continue to play it over and over, day after day, gradually becoming more familiar with it. I may sometimes stop to pull out a problem measure and fix it, but otherwise I'm not preparing it for concert level performance, just playing it for fun, so I just try to get it gradually better.
B. If I'm trying to really get it good, which I've done in the past on occasion, I will:
4. Pull out any problem sections, measures, chords, etc. and work them individually. It always seems like to me every piece has 29 easy measures and 1 impossible measure that prevents me from playing the piece well. And the problem is, if I play the piece 1,000 times, the easy measures get 1,000 times better, but the 1 bad measure never gets any better. So I have to isolate this, fix it, and play it as many times as it takes to get it to the same comfort level as the 20 easy measures, THEN work the piece as a whole.
5. Once I feel like I've "mastered" the technical aspects of the piece (playing the right notes in the right order) I then begin to work on the "artistic" aspects of the piece - dynamics, phrasing, etc.
6. I will have, at some point between 3 and 5, also gotten a feel for things like melody versus accompaniment, or which parts should be foreground, middleground, and background, and so on. So once I've mastered the technical stuff and kind of gotten the general artistic stuff down, I will work on the "nuances" that I feel take it from "playing a piece" to "interpreting a piece" - hopefully in a highly tasteful artistic manner.
Commonly though, I never get past 3A. I just don't have the drive, or need to try to play a piece "at concert level". At this point, it's satisfying enough to be able to get through the piece without it totally falling apart. I often don't play any pieces that have that one too-tricky measure that involves 1,000 times as much work.
So generally I play something I can read through easily, and then hone by playing through a couple of times to make sound not like I'm sight reading it, but have played it a couple of times, then I move on to another piece when I lose interest. Sometimes I work on an entire book's worth of pieces this way.
I'd like to be more "serious" about playing Classical guitar, but to be honest, I'm just not that excited by it. It's kind of like when i go surfing - it's nice to go out there and spend the day catching waves, but I'm not really putting a lot of effort into getting better. I do it for fun, not to challenge myself.