There are two threads of thought that one can find in this period which get at this issue. Neither are the position that all races are just superficial variations — that sort of requires Darwin to really get to that point, much later. Instead one finds either an advocacy of Biblical monogenism as applied to questions of what we would today call race, or one finds the discussions of the ways in which apparent "savages" are in fact culturally equivalent to Europeans.

On the former point, one of the earliest questions that Enlightenment Europeans concerned with taxonomy and geographic diversity asked about human beings was whether they were all essentially the same species or whether they were all different. One sees this is Linnaeus and Buffon and much later in Agassiz. The pre-Darwinian version of this debate was about monogenism versus polygenism, or whether all humans are descended from a single pair of ancestors (as in the Adam and Eve story in Genesis) or whether different geographically distant humans are acts of separate creation. The former tended to be coded as a literally Biblical approach and the latter as a "scientific" approach that understood the Bible was not a literal account, somewhat ironically. There were several 17th century advocates of the idea of monogenism, including Robert Boyle. In general this debate did not, as I understand it, heat up until the 19th century, though (and Darwin's work contributed to this argument — Darwin created a "scientific" monogenic argument).

The other thread is more common in the 18th century: inquiries in the nature of "savages," trying to figure out if they are really that different from "civilized" peoples, or if it was just a matter of what they were used to. One of the key sources of discourage for the Enlightenment were critiques of European societies processed through the lens of external viewpoints (whether indigenous or Asian or what have you). Fitting your timeline, Michel de Montaigne's 1580 essay On Cannibals essentially argues that Amazonian tribesmen were not essentially different, culturally or morally, from 16th-century Europeans who regularly engaged in torture and other practices he found abhorrent. This is not framed in terms of "race" — it is framed in terms of barbarism vs. civilization — but you can see how this debate also ends up having a "biological" component over time. (And, again, became a major issue for later 19th-century theorists like Darwin, who also argued that culture was essentially malleable.)

So the above is not nearly any kind of complete answer, and you are right that such writings are fewer and farther between than the writings of those who found ways to justify European (and American) enslavement, extermination, suppression, etc., of non-Europeans in such topics.

This is an utterly later time period than you are asking about, but Stocking's Victorian Anthropology is an extremely useful resource on pre-Darwinian debates on both of these topics.

Jesus didn't directly taught it, but there's a famous part of Paul's epistle to Galatians, which sounds a bit revolutionary even today:

There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus.

Ancient people didn't think in terms of "races". They thought about peoples, and described their various customs, dress, food, weapons etc. Greeks thought in terms Greeks vs most else (Barbarians). Greeks did think they are in some ways superior to most others. Jews likewise -- they were "the chosen people", everyone else wasn't. (Note that Paul speaks about "Jews" vs "Gentiles", i.e. non-Jews, because it was the most important distinction in that area; in some translations, you'll find "Jew nor Greek").

Rome was a mix of peoples from all sides, including Africa.

Races like homogeneous blocks of "whites", "blacks", "reds" etc don't exist, but it doesn't mean there's no biological variation between various regions and peoples -- for instance, in Europe, where everybody is natively "white", there are significant variations in height, eye and hair color that haven't been really explained. After all, we know that people outside Africa have some small percentage of Neanderthal DNA and that could have some subtle effects...

There was a similar question recently so /u/gynnis-scholasticus has assembled an list of previous posts about the history of racism.

What is missing is that racism seems to be connected to a general belief that superficial physical characteristics must be connected to everything about the person. For example, phrenology was an attempt to connect bumps on the skull to various mental characteristics. There was also a lot of 19th century literature where shape of nose, chin, relative length of fingers etc was supposedly connected to much else. People with red hair were often associated with violence or negative characteristics. There seems to be an obsession with measurable physical characteristics peaking in the 19th century.