The AskHistorians reading list is a good place to start. There's a page on Middle Eastern history, including Mesopotamia and Persia, and a section on Greek history.

In addition to those on the booklist, for the Sasanian Empire, Touraj Daryaee's works are great. I've heard it said from experts that "From Oxus to Euphrates" is the best intro.

If you're interested in Persian religion, I suggest Boyce's Zoroastrians: Their Religious Beliefs and Practices, which traces out its history from before Zoroaster to the present day.