Well, the Covenant was in a bit of a "mad dash" to locate the Halo Rings as well as the Ark in order to begin their "Great Journey". The problem with this is that the Ark that they had located was on Earth, so the UNSC needed to defend it from the Covenant invasion forces, who, might I add, didn't originally intend for there to be humans there. Another area where the Covenant's goals conflicted with that of the humans is the way that firing the Halo Rings is simply a way of wiping all life outside of the Arks from the Galaxy. This turned it into a battle of survival for the humans - defeat the Covenant, or be vaporized by the Rings.

The Forerunners intended for the Rings to be used to wipe out the Flood, a parasitic race that consumes entire races. The Forerunners had been fighting a losing battle against the Flood, and the Halo Rings were a sort of final option to save the galaxy by wiping out everything except certain amounts of each race put into Arks to carry on life. Since the Flood was discovered to have survived on the remaining Rings and Installations left by the Forerunners, the Covenant was fulfilling the Forerunners intents in a way by trying to blow the Rings, which would wipe out the Flood. The place they strayed from the Forerunners intent was by intending only to save the Covenant, and not the rest of the life in the Galaxy.

I hope this helped.

Sources: I played the games, read the books, amd watched Halo Legends.