The first dialog we are reading is titled, in the original Greek, either Euthyphro or peri hosion (i.e., about "hosion").
"hosion," can best be translated as "holiness" or perhaps "righteousness" or maybe even (very loosely) as "justice" or "goodness."
Thus, this dialogic conversation is about what it means to be a holy or good or righteous person.
In the Greek, "hosion" clearly has the connotations of both moral rectitude (doing the right thing) and of being properly religious and spiritual. Euthyphro, of course, believes he is both. The dialog is about what that means.