The question is asked too broadly and with not enough context to answer it succinctly. I will try to give a few general pointers, being well aware that my answers will maybe not cover what @forkandwait had in mind. If you want more direct answers then ask more direct questions.
There are three basic dimensions of a music piece. These can be analysed separately although they are of course deeply connected to each other. (Separating them is just a device to make the analysis easier, but always keep in mind that at some point you need to come to interconnected conclusions from the results of these separated analyses.)
- rhythm and rhythmic development
- melody and melodic development
- overall form or structure
Rhythm is a fundamental quality of any musical piece. Quite often it is connected to certain dances and their characteristic movement patterns (waltzes, gavottes, sarabandes, ...). (West-)African and west-african-rooted music makes even more complex use of rhytmic patterns (polyrhythms, cross-rhythms).
Melodic development is the counterpart of rhythm. There is a static aspect of it (harmony) and a dynamic aspect (counterpoint in modern music, basso continuo in the baroque era, etc.). Notice, that most (printed) music theory is ultimately based on the european music tradition and uses the terminology and methods of it. This is based on physics, mind you, but i.e. arabic music makes use of more partial harmonics than european music ("quarter notes"). So, depending what you analyse, it might pay off to add other than european music theory to your background. I.e. analysing indian ragas in terms of european harmonics will not likely lead to results that make sense.
Overall form is also heavily dependent on what you analyse. For european classical music there is i.e. the sonata form, the Cantata, etc., all these based on older forms like the concierto grosso, the (from here on no more links as i can't post them) chorale, the madrigal, etc.. Notice that other musical traditions have different forms - i already mentioned indian ragas - and, like with harmony, you need to learn about the background of the respective cultural environment before you can adequately analyse a piece from that tradition.
Edit: a few sources for further study, albeit all centered on european classical music and in German:
Gradus ad Parnassum (J Fux); the father of them all
Handbuch der Harmonielehre (H. Riemann); Function Theory
Synthetische Harmonielehre (F. Neumann)
Die Natur der Harmonik und Metrik (M. Hauptmann)