Depends on the piece and the level of analysis. Who's the intended audience? But generally, it's good to start by noting the most prevalent keys, their relationships, and the piece's overall form. From there you can go deeper and analyze the harmonic progressions using e.g. Roman numerals representing chord functions, and this will help you spot key modulations and understand their role in the piece. If you really want to get to know some music, identify its themes and motives and find where they pop up later, how they may have been modified from the originals (in a different voice? melodically inverted? chromatically altered? retrograde?), etc.
And now that I've said all that, don't take it too seriously.
The point of analysis, IMO, is to find patterns, because artful music is almost always filled with such. The above are just a few of the kinds of patterns which can be found. Keep them in mind when you're analyzing, by all means, but don't be constrained by them. If you spot something interesting that doesn't have a handy entry in the musical dictionary, it's probably even more interesting than the patterns that do.