Were people who followed minority beliefs actively persecuted, tolerated but with lesser legal rights, or completely accepted, i.e. today's standards of religious acceptance. How did this change throughout the centuries, and why? Were any beliefs persecuted because of a single specific event, or was a standard of hatred maintained simply because 'We don't like those people'?
I apologize in advance for the excessively broad question.
Each cult was different, but you might be interested in one in particular. In the mid 2nd Century AD a snake cult developed in northern Anatolia. It was led by a charismatic, shady character called Alexander of Abonoteichus.
This Alexander supposedly had a background of being a confidence man or huckster, and he travelled a bit around the Roman Greek East. He eventually set up in a town called Abonoteichus (later Ionopolis) in the north of Asia Minor and set up a snake cult.
He gave prophecies and set up a temple and the cult became fairly popular. We know it spread a bit because a statue of the snake has been found as well as inscriptions to avert the plague which we know came from Alexander.
The interesting part is that a writer called Lucian wrote a detailed history of the prophet and the cult. Lucian was a hostile writer. He claimed Alexander was a fraud (even the title of the work is sometimes called Alexander the False Prophet) and that his followers were all stupid country bumpkins. As a piece of satiric literature it is quite good.
However, this cult was not actively persecuted by the authorities. In fact, there are hints that it received some imperial favour, like personal connections between Alexander’s and imperial staff members, as well as a prophecy given to the emperor Marcus Aurelius during a war.
Another interesting bit of Lucian's work is that it describes how Alexander and his cult reacted towards other cults - namely Christians, the Epicurean philosophers, and unknown 'atheists.' Apparently Alexander didn't like competition at all, and went out of his way to attack them.
Some cults were persecuted and some were not. It depends entirely upon the situation. Marcus, for example, seemed to have been rather tolerant. At other times the authorities were pretty hard on the cults. The best known example is the persecution of Christianity at various times and places. At other times the emperors were lenient on Christians. Check out the letters of Pliny and Trajan for an example.
But if you would like to read a primary source on a minority cult, you might check out Lucian’s Alexander the False Prophet .