In these opening lines, Oedipus is referring to the old priests and the younger citizens of Thebes who have come to him as suppliants; this is why he describes them as "young and old." Cadmus was the mythical founders of Thebes, and thus the phrase "sons of Old Cadmus" is just another an ornate way of saying "Thebans."
Cadmus the mythical founder of Thebes was a prince, son of the Phoenician king Agenor and queen Telephassa of the fabulously wealthy trading city of Tyre. Cadmus was sent to find his sister Europa. He consulted the oracle at Delphi which told him to follow a cow with a distinctive half-moon marking and build a city where the cow lay down.
Cadmus had to kill a sacred dragon guarding a nearby spring. Athena told Cadmus to sow the dragon's teeth in the ground. From these teeth sprang up warriors who, along with the sons of Cadmus and Harmonia, became the ancestors of the noble families of Thebes.