We know that there were black and white schools, bathrooms, restaurants, seats on the bus, etc., but what happened to the Asian-Americans, Latino-Americans, and other people of color at this time?
I can only speak for Florida on this one, but the 1900 Florida State Statutes book specifically states "negro", and defines a negro to be a person of 1/8 negro heritage or more. Then it goes on to list laws specific to them. It doesn't mention any other peoples.
One terminology correction: "Separate but equal" comes from a Supreme Court case. It wasn't "passed," it was "decided" in Plessy v. Ferguson, 163 U.S. 537 (1896). How states drafted, enacted, and implemented laws based on that doctrine is what you're asking.
Most of the famous case law on "separate but equal" and desegregation is, of course, on black/white race relations. However, I'm aware of at least one case, Lum v. Rice, 275 U.S. 78 (1927), that applied the Plessy doctrine to uphold a state decision keeping a Chinese girl out of white schools. That decision was impliedly overruled with Brown, but not specifically abrogated until 1971, when the federal courts directed the desegregation and opening of English language opportunities to Chinese immigrants in San Francisco. I'm afraid I can't speak to social history, but this may be a starting point for your own research.