From what I gather once the people historians call Bulgars came to the Balkan peninsula, and following their victories against the Romans, they began to rule half of the southern Slavs. These would be the Bulgarians. The other half, which were not conquered or not assimilated enough to remain, would be the Serbians.

In that sense, Bulgaria predates Serbia, because Serbia is defined as not-Bulgaria, but its a semantic argument. The two identities were forged at the same time, by the people historians call Bulgars, based on how much land they could hold and how far west they got on the peninsula. The two nations were also formed at about the same time, as the same reaction against the weakening Ottoman empire and in the context of pan-Slavism and the rise of Russian imperials strength and the plans for the straits.

In many ways this divide is artificial, and historically much of Bulgaria and Serbia have been part of the same political unit (be it the Roman empire, or Bulgaria, or the Ottoman empire) for longer than they have been divided. Hopefully they are rejoined in the European Union or its successor organization soon, and stay together.