What is the conflict that starts the problems between Agamemnon and Achilles in the first place?
This is a good question and one that is essential to explore, if you are going to understand the work. Here is the context:
After some victory among the Greeks, both Agamemnon and Achilles were awarded prizes (women as part of the spoils of war). Agamemnon obtained a woman named, Chryseis, and Achilles obtained a woman named, Briseis. However, Agamemnon had to give his "prize" back, because she was the daughter of the priest of Apollo. The god, Apollo, was attacking the Greeks with great fury and the only way to stop this was to give back the daughter of the priest of Apollo.
When this happend, Agamemnon wanted to take Briseis away from Achilles. This act was due to his pride, but more importantly it slighted the pride and honor of Achilles, which was paramout in Greek culture. For this reason, Achilles was filled with rage and decided not to fight for the Greeks. This caused the balance of power to favor the Trojans.
In terms of the rage of Achilles, here is what Homer says:
"Rage—Goddess, sing the rage of Peleus’ son Achilles,
murderous, doomed, that cost the Achaeans countless losses,
hurling down to the House of Death so many sturdy souls,
great fighters’ souls, but made their bodies carrion,
feasts for the dogs and birds,
and the will of Zeus was moving toward its end.
Begin, Muse, when the two first broke and clashed,
Agamemnon lord of men and brilliant Achilles."
In many ways, the Iliadis really about the rage of Achilles.