There were trade routes between Egypt and Mesopotamia by the fourth millennium BCE, and I discussed the archaeological evidence for this in Mesopotamians and sumerians. Unfortunately, our knowledge of Egyptian-Sumerian relations is virtually entirely based on archaeological evidence.

Mesopotamia did not really appear prominently in Egyptian texts until the latter half of the second millennium BCE, and we know a fair amount about Egyptian and Babylonian relations. There's always more to be said on the topic, but I wrote about this in Is there any proof that the Babylonians and Egyptians had contact with each other and if they did, what was their relationship like?