No, the answer doesn't begin with the fact that Iranian revolutionaries took American diplomats hostage during the Iranian revolution in 1979, which is what the mainstream press often asserts. The U.S. hatred for Iran stretches back even further than that.
In fact, to understand the hatred that the Pentagon, the CIA, and the NSA have for Iran, it is actually necessary to go back to the founding of the United States.
When the delegates met at the Constitution Convention, it was for the purpose of modifying the Articles of Confederation, a type of governmental system under which Americans had been operating for 13 years. Under the Articles, the federal government's powers were so weak that it didn't even have the power to tax.