Obviously, the Achaemenid empire was quite cosmopolitan due to it's size and importance, comprising of many religions and ethnic groups. Often, however, it is said that Zoroastrianism was the state religion, but I've heard recently that the vast majority of people, including those in the court, followed a proto-Zoroastrianism or rather the religion that Zoroaster reformed, while what we think of as Zoroastrianism was a minor movement or more of an intellectual movement. First, is this claim true? Second, to what extent was it the state religion? Finally, third, what did Zoroastrianism look like to the people of the Achaemenid Empire?
If I were to answer your questions in the strictest sense, the answer would pretty much be "nobody knows". Sources on the Achaemenid Empire are sparse enough to begin with, and ones that usefully discuss their religion essentially non-existent. But before we start, there's another thing: please forget virtually everything you ever read about Zoroastrianism on the internet. There is an absurd degree of misinformation and grossly outdated scholarship on it in places like Wikipedia.
To get a sense for a large part of the source on this that we do have, a good place to start is the Achaemenid royal inscriptions, available on the ever helpful livius.org: http://www.livius.org/sources/content/achaemenid-royal-inscriptions/ Now you may want to read the Behistun inscription in its entirety: http://www.livius.org/sources/content/behistun-persian-text/ But probably the most useful for our purposes is the XPh inscription. http://www.livius.org/sources/content/achaemenid-royal-inscriptions/xph/
A great god is Ahuramazda, who created this earth, who created yonder sky, who created man, who created happiness for man, who made Xerxes king, one king of many, one lord of many.
I am Xerxes, the Great King, king of kings, king of countries containing many kinds of men, king in this great earth far and wide, son of king Darius, an Achaemenian, a Persian, son of a Persian, an Aryan, of Aryan stock.
It is at least rather apparent that Xerxes venerates (or found it appropriate to portray himself as venerating) Ahura Mazda as, if not a "supreme being", then at the very least as a creator deity. Something that is less apparent in translation: There is a formulaic use of a particular word for "great" (vazraka) here: Ahura Mazda is the great god, Xerxes is the Great King, ruler on the great Earth. The text gets particularly interesting later on:
King Xerxes says: when I became king, there was among these countries one that was in rebellion. Ahuramazda bore me aid. By the grace of Ahuramazda I smote that country and put it down in its place.
And among these countries there was a place where previously demons (daiva) were worshipped. Afterwards, by the grace of Ahuramazda I destroyed that sanctuary of demons, and I proclaimed: 'The demons shall not be worshipped!' Where previously the demons were worshipped, there I worshipped Ahuramazda at the proper time and in the proper manner. And there was other business that had been done ill. That I made good. That which I did, all I did by the grace of Ahuramazda. Ahuramazda bore me aid until I completed the work.
You who may live hereafter, if you should think 'Happy may I be when living, and when dead may I be blessed,' have respect for that law which Ahuramazda has established. Worship Ahuramazda at the proper time and in the proper manner. The man who has respect for that law that Ahuramazda has established and worships Ahuramazda at the proper time and in the proper manner, he both becomes happy while alive and becomes blessed when dead.
The two first paragraphs are especially interesting in conjunction. Since Xerxes starts out by discussing a rebellion, we can infer that "a place where daiva were worshipped" was not merely a euphemism for a rebellious province or the like. While the text is woefully lacking in detail, we can at least tell (given that the Achaaemenids did not force everyone to adhere to Zoroastrianism) that there was some sense of "improper practice" serious enough for the Great King himself to intervene. Was this a general feature of Achaemenid kings, or was Xerxes simply especially pious? We just can't really know.
In another inscription we find:
Artaxerxes, the great king, the king of kings, the king of all nations, the king of this world, the son of king Darius [II Nothus], Darius the son of king Artaxerxes [I Makrocheir], Artaxerxes the son of king Xerxes, Xerxes the son of king Darius, Darius the son of Hystaspes, the Achaemenid, says: My ancestor Darius [I the Great] made this audience hall [apadana], but during the reign of my grandfather Artaxerxes, it was burnt down; but, by the grace of Ahuramazda, Anahita, and Mithra, I reconstructed this audience hall.
Here we find Artaxerxes invoking three deities: Ahura Mazda, Anahita (a water deity), and Mithra (can perhaps be summed up as a judicial deity or entity, responsible for oathkeeping, covenants, etc).
The sacred scriptures of Zoroastrianism, the Avesta, would only be compiled nearly a milennium after this time, most likely selectively so, by the Sasanians. This is rather frustrating, because it means that we don't know what traditions the Achaemenid kings are speaking from when they interrupt their self-aggrandizement to make the occasional interesting comment. Even by the time of the Sasanians, it is interesting to note that in parallel Greek inscriptions, we find "Ahura Mazda" replaces by "Zeus" - perhaps a functional equivalence rather than acknowledgement of any deeper connection, but interesting nonetheless.
For further reading, the go-to author for better understanding of Zoroastrianism and its history remains Mary Boyce.