Before us is Mishnah 7 in the third chapter of Maseches Terumos. In the previous Mishnah we learned that there is a fixed and binding order to the separations: first bikkurim, then Terumah, and finally maaser. Our Mishnah comes to clarify the source of this order and the reason behind it.
How do we know that bikkurim come before Terumah:
The Mishnah asks: "U'minayin sheyikdemu bikkurim laterumah" - what is the source in the verses that bikkurim must come before Terumah? After all, both of them are described in the verses by the very same two titles, as the Mishnah puts it: "Zeh karui terumah v'reishis v'zeh karui terumah v'reishis". 'Terumah' means the portion that is lifted off and separated, and 'reishis' means the first, and both titles apply equally to Terumah and to bikkurim alike. If so, how do we know that bikkurim come first?
The Mishnah answers: "Ela yikdemu bikkurim shehen bikkurim lakol" - the reason bikkurim come first lies in their very name. Bikkurim are the first fruits, and just as the word 'bechor' denotes the one born first, so too the bikkurim are the first fruits that come into a person's hands. And since they precede everything else, it stands to reason that they should also be separated before all the other separations.
Why Terumah comes before maaser rishon:
"U'terumah larishon, shehi reishis" - how do we know that separating Terumah gedolah for the kohen comes before separating maaser rishon, which is given to the Levi? Because Terumah is called 'reishis' in the Torah, whereas maaser rishon is not called by that name. This is what the Mishnah means when it says "shehi reishis": Terumah is called reishis, and maaser rishon is not given that name in the Torah.
Why maaser rishon comes before maaser sheni:
"V'rishon lasheni, sheyesh bo reishis" - how do we know that maaser rishon, which is given to the Levi, comes before maaser sheni, which remains in the owner's hands? The Mishnah answers that this follows from logic: within maaser rishon itself there is something called reishis - namely terumas maaser, which likewise must go out to the kohen, and which is likewise called Terumah just as Terumah gedolah is called Terumah and reishis. And since maaser rishon contains within it something of reishis and something of Terumah, it stands to reason that it should come before maaser sheni, which contains nothing of reishis at all.
In summary: In this Mishnah we learned the source and the reason for the order of the separations. Bikkurim come before Terumah, even though both are called Terumah and reishis, because they are the first of everything. Terumah comes before maaser rishon, because it alone is called reishis. And maaser rishon comes before maaser sheni, because it contains within it something of reishis - terumas maaser - whereas maaser sheni contains nothing of reishis at all.