What were some of Martin Luther's and John Calvin's criticisms of religious beliefs and practices?
This is a huge question. For example, Martin Luther posted 95 theses! This shows that his issues were not a few. In light of this, let me mention three of his criticisms.
First, Martin Luther questioned the selling of indulgences. The jingle that was popular in his day was: "As soon as a coin in the coffer rings, a soul from purgatory springs." Luther saw this as corrupt and against the teachings of the New Testament. Here is a quote:
“Christians should be taught that he who sees someone needy but looks past him, and buys an indulgence instead, receives not the pope’s remission but God’s wrath.”
Luther also questioned the building of St. Peter's Basilica. He reasoned:
Why does not the pope, whose wealth today is greater than the wealth of the richest Crassus, build the basilica of St. Peter with his own money rather than with the money of poor believers?
Others shared this sentiment as well in his day and age.
Luther also questioned the pope as the highest authority of the church. As he looked at history, he knew that popes made mistakes in the past and could do so in the future. For him, only Christ was the head of the church. Hence, he questioned the authority of the structure of the church.
As for Calvin, he agreed with Luther on these points, but we can add a few more criticisms.
First, Calvin, as a classical humanist in training, believed in the importance of going back to the sources (ad fontes). He wanted to go back to the early teachings of the Christian church, rather than be muddled in the history of medieval scholarship. Tradition was important to him, but not as important as Scripture.
He also rejected the veneration of Mary. He, of course, viewed her highly, but saw the Catholic veneration of Mary as unfounded in Scripture, which included the immaculate conception.
Arguably the greatest criticism of both John Calvin and Martin Luther was on justification (salvation). They rejected the Catholic sacerdotal soteriology that believed that grace was infused at baptism to enable a person to cooperate with grace to work out his or her salvation. For them, justification was by faith alone, apart from works. Calvin famously said:
"For we maintain that a man is justified by faith, but that faith which justifies is never alone."