I am looking for any hints or experience reports or materials/potential difficulties about how to introduce the use of Greek letters in high school Math/Physics.
My main point of advice (based on teaching maths in engineering contexts, either at or post-high school) is to introduce these new letters properly, rather than try to glaze over them.
Take for example the equation for resistance, R=ρL/A. As well as defining what each symbol represents (e.g. ρ is resistivity), I name the greek letter verbally, and write its name in Latin letters alongside the equation ("rho") when I first introduce it. I typically mention that it's a Greek letter, and if there's an alliterative link that can add meaning or aid recall (such as ρ -> r -> resistivity) I'll highlight that too.
Students become familiar with the new symbol, fairly quickly but don't remember its name immediately after hearing it spoken once; the ability to look back at their notes and read the name helps a lot in the early sessions. I continue to refer to the symbol by name to aid recall, but don't write out "rho" again after the first introduction.
There is an MSE question very similar to this. (Closed now, but with good discussions, answers.)
Also, some lesson plans and the like if you just Google.
I didn't have a huge problem with it, just dealt with them as they came up. But am a strong student overall. All that said, I still don't really know them all, just the ones I've encountered.
I think a fun memorization (flashcard practice, tested) sidetrack to all the math/math would be to cover it as a lesson for a day's hour (with a short quiz, next day). And it may get a few of the arty, humanities types interested and enjoying something off the math/math track. But just when doing it, let the kids know that this will be slightly different from all their calculational practice so that they can reframe themeselves.
I think the right spot/time is in "Algebra 2" (US meaning). They will have maybe seen a few already, maybe in chemistry or geometry. This now gives them a chance to learn all the primates in the zoo.
I wouldn't bother with it in a physics or chem class though. Just use the few you need and clarify the meaning, pronunciation, "false friends" (letters that look same as English but are not, etc.)
And "concept lovers" will scoff, but there is some enjoyment in learning things like this. (Lists of state capitals are much enjoyed by small children...not a sin.)
I would restrict to using it rather conventionally (alpha, beta, gamma for angles, lambda for wavelength, rho density, etc.). I would not introduce the confusion of using a symbol in an abnormal fashion. And I would not stick in Greek when not needed (or at least not normally used). I hate the use of Fraktur, e.g. I would avoid the use of "squiggle" and "other squiggle" unless really needed (unlikely in high school classes).
If their book has a list of them (many do, where the formulas are listed), point it out to them. If not, give them a paper handout.
P.s. Answer to question in first comment: the first is "squiggle", the second is "other squiggle". This was actually semi-allowed in Navy Nuclear Power School, to refer to the Greek letters that way. Squiggle is a common sign for logarithmic energy decrement, which comes up in the context of collisions during neutron thermalization. I can't remember what other squiggle did, maybe delayed neutron precursor fraction or fast fission. Something of that sort. If you do a Google search, it's not just a Navyism, but lots of engineers and physicists refer to the Greek letters that way. Also, at NPS, you were also allowed to refer to Y(0) and J(0) Bessel functions as Yo and Jo. Like "Yo, Adrian". Little bit of grain for the hard working kine...