> Here's how I approach it: forget about all the historic naming like "perfect fifths" and just think in terms of the modern 12-note equal temperament.
You actually want to learn these names, because they characterize the diatonic scale. Basically, pick seven contiguous notes on the circle of fifths, you get the pitch-class set of a diatonic scale. Then put the notes in the set in pitch order, and pick a tonal center. The default, naïve choice (pick the "flattest" note as your tonal center) is called Lydian mode. It can be interesting, but it has a drawback in that it forgoes the subdominant relationship. Picking the next-to-'flattest' note gives you Ionian, which solves this (the fourth scale degree forms a "perfect fourth" with the tonic, which is the flip side of a perfect fifth. Having more notes that can be related to the tonic makes for more musical possibilities). This is one simple explanation of the diatonic scale we ordinarily use.
One other quirk that also explains the "weird historic names": in traditional music theory, sharps and flats are definitely not treated equally, the way that would be implied by 12-equal temperament. The musically-relevant distinction is simple enough to explain: taking one example, F# "wants" to step up to G, whereas Gb "wants" to step down to F. This means that the "circle of fifths" turns into more of a helix of sorts that can be extended in both directions, in principle indefinitely. This difference cannot be "heard" directly; it's all about characterizing how the notes "work" in a piece of music.
After learning about the diatonic scale degrees, the next sensible step would be to start learning about counterpoint, which is based on simple definitions of consonance and dissonance between scale degrees. Then move on to thoroughbass and harmony.
You actually want to learn these names, because they characterize the diatonic scale. Basically, pick seven contiguous notes on the circle of fifths, you get the pitch-class set of a diatonic scale. Then put the notes in the set in pitch order, and pick a tonal center. The default, naïve choice (pick the "flattest" note as your tonal center) is called Lydian mode. It can be interesting, but it has a drawback in that it forgoes the subdominant relationship. Picking the next-to-'flattest' note gives you Ionian, which solves this (the fourth scale degree forms a "perfect fourth" with the tonic, which is the flip side of a perfect fifth. Having more notes that can be related to the tonic makes for more musical possibilities). This is one simple explanation of the diatonic scale we ordinarily use.
One other quirk that also explains the "weird historic names": in traditional music theory, sharps and flats are definitely not treated equally, the way that would be implied by 12-equal temperament. The musically-relevant distinction is simple enough to explain: taking one example, F# "wants" to step up to G, whereas Gb "wants" to step down to F. This means that the "circle of fifths" turns into more of a helix of sorts that can be extended in both directions, in principle indefinitely. This difference cannot be "heard" directly; it's all about characterizing how the notes "work" in a piece of music.
After learning about the diatonic scale degrees, the next sensible step would be to start learning about counterpoint, which is based on simple definitions of consonance and dissonance between scale degrees. Then move on to thoroughbass and harmony.