Or with the discovery of the new world came supremacy based on race?
The only thing i can think of is between local tribes.
Yes, very much so. We can see the beginning of a proto-racist culture (as opposed to jingoism say between Sparta and Athens) in classical antiquity.
The idea of superior and inferior peoples is explored especially well by this book by Benjamin Isaac. Check the first chapter out, most of it is available there.
You can see in some authors like Herodotus that the Greeks were already associating certain physical features with the regions in which they were found, and had ways to explain it.
Consequently, they tried to associate personality traits in a similar way, with Aristotle famously arguing nomads were the "idlest of peoples."
The way in which these traits were analyzed certainly led to a kind of proto-racism in which the notion of the superiority of certain peoples was being explored, and often advocated. Still, we must accept that this appears to be a view that is still establishing itself, and it relatively subtle and could well be scientific in its endeavours. Whether it carries the same kind of malice as, say, biological works on black people during the peak use of slavery in the US, is doubtful, though.
Is your question specifically asking about the people of the Americas? Because there were certainly interactions between different racial groups in Africa, Asia and Europe before that point.