Short answer: the best way to learn something big and complex is to make sure you start at the beginning, and learn in small steps that build on each other.
Here's what I would do if you came to me as a private student. I'd figure out what your goals are—it sounds to me like your primary goal is to appreciate and better understand major works of the canon of Western, tonal, so-called "classical" music. You've recognized that you can enjoy a piece better when you study it, and you're very interested in the form and structure of pieces.
So the main focus of the study I'd assign you would be music theory. But I would definitely start by teaching you to read music, and I'd use an instrument to help you learn. I would probably encourage you to use piano, just because it's the most useful as you continue studying music theory. If you wanted to explore other instruments that are "easy to learn," then even a recorder or ocarina could get you started. To one of your core questions: Yes, you could "learn to read music" without ever playing an instrument at all, but it would be a bit like learning to read a written language without ever speaking. It's possible, but it's much easier to put it into practice, just because it will engage experiential pathways in your brain rather than purely conceptual ones. You certainly don't have to learn to play well, or quickly, or at anything more than a beginner level. I can't play anything on piano up to tempo that's much harder than a scale, but I still find it helpful to do music theory analysis sitting in front of a keyboard.
The most important thing is to "scaffold" your learning. I would advise you to get the starting book of a beginning piano method; this will probably start with some tunes that use only two or three notes. As you play them, you'll get familiar with the placement on the staff for those pitches. The beginning tunes will probably use all notes of the same length, too; after expanding your range to 8 or so notes, they might start varying the rhythm with longer or shorter notes. You "learn to read music" the same way you learn the instrument: by introducing only a few concepts to yourself at a time.
In fact, music theory study is similar. I can't teach you about German sixth chords before you know what chords are in the first place, and the definition of a triad, and a sixth, and that means talking about intervals, and that means talking about pitches. Similarly, analyzing the form of a sonata-allegro movement often comes down, not to recognizing melodies, but to identifying changes of key (or "tonal areas,") so that means understanding key, tonality, tonicism, tonic and dominant chords, cadences, etc.
Now, to answer the "question you didn't ask": If your end goal here is not to play an instrument or know music theory, but to enjoy music more by understanding it better: there are other kinds of study you can explore as well. To appreciate a musical artifact, it certainly helps to understand the context that created it. I would recommend exploring a course on the history of Western music. As you start at the beginning with ancient and medieval practices, this will open your ears to systems of music-making that are as far removed from Mozart as Charles Ives is, but as you proceed through time, by the time you get back to Mozart you'll understand how he was informed by Haydn, who was informed by Bach, and so on. Meanwhile, I'd encourage you to study the musics of other cultures too. You might not enjoy all of them as much as "classical" music, but they help make you aware of conventions within Western music that you might have mistaken for universal aspects of music, but are in fact cultural (like "having melody and harmony," "being in a key," and dividing the octave into 12 half steps). Being made aware of the phenomena, you're better able to appreciate them in Western music rather than take them for granted.