In parshah Tetzaveh, a ceremony is prescribed for “dedicating” the Kohen Gadol (High Priest) and his sons for service in the Mishkan (later: in the Temple). In this case, the “High Priest’ is Aharon, and the sons are Elazar, Itamar, Nadav and Avihu. A grandson, Pinchas, unmentioned here, later becomes Kohen Gadol at the passing of his father, Elazar.
This ceremony takes place only once. It is not a general rite for all kohanim in every generation. By initiating Aharon and his sons, the Israelite priesthood itself is initiated.
This ritual “sanctifies” the individuals and their descendants. Mosheh is told:
“This is what you shall do to sanctify [לקדש] them…” 
ֵElsewhere in the same parshah, Mosheh is told that he’s “ordaining” them:
“You shall then ordain [מלא] Aharon and his sons…” 
The first part of the initiation was a “sin-offering.”
The Chatat was offered for unintentional sins, not for those done deliberately. It was to be assumed that if Aharon or any of his sons had performed a deliberate sin, they would have already confessed, done teshuvah and made restitution if possible.
Plus, the Mishkan not yet being in use, how could they have done even unintentional sins regarding sacrificial procedures and/or utensils?
If we look at the offerings as reflecting required attitudes, not just as mechanical procedures, we might see their purpose more clearly.
Had there been an unintentional or unknown “error” in their preparations, not in their actual service in the Miskan, as that service had not yet begun? The kohanim, before being initiated, approach the ceremony willing to admit that despite their own best efforts, they could not fully vouch for their own perfect preparedness. Only G-d could do that.
I think that the “sin-offering” here reflected an attitude of humility, as if to say, “We’ve prepared ourselves in every way we know, but if we’re still responsible for errors that we might have made, even if unknown to us, we ask forgiveness and accept G-d’s Will.” The nazir had similarly to bring a sin-offering as he ended his period of abstention — in case he had inadvertently violated it in some way. To assume that we are perfect because of our own efforts is arrogant. Rather, it seems to me that the kohanim entered the priesthood by saying they accept Divine Judgement on their readiness.
The ritual then moves to the sacrifice of not one, but two rams. “Two” does not here mean an intensified attitude of devotion.
Why two rams?
I don’t believe they represent separate ceremonies, as much as two parts of a single rite:
The first ram serves as a “burnt offering.” The burnt offering is always associated with communing with G-d, represented in Jewish liturgy by the Amidah — the central prayer of every service. Its blood is “dashed” or “sprinkled” on the sides of the altar, uniting the life that is in the blood — now identified as representing Aharon and his sons — with G-d’s life (the altar representing not G-d, but the act of communing with G-d). 
From the second ram that was sacrificed, Mosheh takes some of the blood, places some on each priest’s right ear, right thumb and right big toe, and sprinkles the rest of the blood around the altar. He then takes some of that blood from the altar, mixes it with anointing oil, and sprinkles the priests’ clothing. 
While these directions are found in Sh’mot/Exodus, they are later enacted in Vayikra/Leviticus 8.
The second ram’s blood, placed on each priests’ ear/thumb/toe, signifies a sharing of the holiness of the altar with the entire person — “top to bottom,” as it were.
Gathered from the altar and sprinkled on the priests’ garments, it demonstrates a special connection between G-d and the priesthood, especially when they are acting in the priest’s role.
Doesn’t the burnt offering itself accomplish this unification?
Yes, but the symbolism here represents not just the individual priests, but the priesthood itself.
What attitude does it suggest that a priest have?
Perhaps that when serving as priest, he represents not himself, but the entire tradition of serving as one who must be prepared to bring harmony between people and G-d.
It’s no wonder, then, that prophecy in ancient Israel began with Sh’mu’el — a child who had been dedicated to, and grew up in the Mishkan:
“[He] was the last of the Hebrew Judges and the first of the major prophets who began to prophesy inside the Land of Israel.” 
Prophecy partook of the holiness of the Mishkan.
But it was Sh’mu’el first, then the later prophets, who brought it out from the Mishkan into the wider world.
 Sh’mot/Ex. 29:1
 ibid. 29:10
Plaut points out that [מלא] is related to a verb that means “to fill,” and suggests that it idiomatically refers to the initiate “filling his hand(s)” with the necessary offerings.”
 ibid 29:16
 ibid 29:21