One who offers
The basic definition of a priest is “one who offers”. So we had better start with Cain and Abel, who both offered to God [Genesis 4:2-8].

Cain and Abel, Titian
Other civilisations did other things with their offerings, like burying them in the ground or throwing them in the sea. The Hebrews believed their God to be ‘up there’, and thus offered burnt offerings so that the smoke rose. After the Flood, Noah did the same [Genesis 8:21-22], and God smelt the pleasing smell. It is regrettable for vegetarians, but according to this understanding God really liked his steak.
But note that there is no person to make the offering for them. They do it themselves.
The first Biblical person to be named a “priest” is the mysterious Melchizedek [Genesis 14:18-20]. When Abraham had won a military victory – do we see Abraham like this? – Melchizedek, who was also “king of Salem” met him, blessed him and gave him bread and wine. In return, Abraham offered him a tenth [tithe] of his booty. The odd thing is that Melchizedek, though a “priest” is not offering to God. He is giving Abraham a form of welcome (bread and wine) and a blessing. And Abraham offers him a kind of tribute.
Whatever it was that made Melchizedek a “priest”, it does not seem to be what we might expect.