Has Catholicism changed it's Teachings?
The Catholic Church has never fallen into error and she has never changed her teachings because the Catholic Church is God's and He is Truth itself and so she cannot teach error. To suggest that the Catholic Church could teach error would mean that Jesus, the Founder of the Church, lied when He promised to be with His Church "always until the end of the world" (Matt 28:20), and to ask the Father to send the Holy Spirit of "truth" to be with us always (Jn 14: 16-17). Since Jesus cannot lie, His Church cannot teach error.
"The Catholic Church cannot be infallible because it has changed its teachings."
The Church teaches all that Christ taught, whether implicit or explicit. Essentially she exists just as He would have her exist. There may have been many secondary developments through the ages, but they were all foreseen and approved by Christ. After all, Christ established a living Church, and a living Church grows. He likened it to a seed. Even as a boy grows into a man with exactly the same personality, yet with many secondary changes in size and knowledge, so too has the Church rightly developed. Christ is God. Christ is the Truth. Christ is the Head of the Church which is His Body.
"What about the Church's approval of slavery?"
The Catholic Church's stance toward racial slavery has always been condemnatory. Though there are different forms of slavery, and though many Catholics over the centuries have been involved in it, the Popes have always vigorously condemned the forcible enslaving of innocent human lives. While the Apostles did indeed tolerate the ancient form of slavery, they didn't necessarily approve of the practice. In fact, St Paul hints that Philemon should free his slave (Philemon: verse 21).
"What about the Church changing its teachings on usury?"
The Church's basic teaching in this area never changed. It still condemns the charging of excessive interest for money loaned (usury). However, as economic systems changed from barter economies to economies based on money, which could increase in value, the Church recognised a distinction between a lender taking exorbitant interest for a loan (usury) and taking a fair return for the risk involved in making a loan.
"Prove that the Church did not fall into error when it sold indulgences."
The abuses relating to indulgences at the time of Martin Luther involved the meritorious work of almsgiving for the construction of St Peter's Basilica (a worthy condition for the granting of an indulgence); they were never sanctioned by the Church but, on the surface, one could gain that impression (hence the reforms of the Council of Trent & Pius V abolishing all grants of indulgences in return for alms).
"The Catholic Church has invented new doctrines."
The so-called invention of new doctrines, which refers to the Church's proclamation of new dogmas is utterly baseless. "New" dogmas of the Church are actually old doctrines dating back to the beginning of Christianity. In proclaiming them to be dogmas, the Church merely emphasised their importance to the Faith and affirmed that they are integral to God's revelation to mankind. The Church followed the same procedure in the 4th century when the New Testament was proclaimed to be divinely revealed.
"The laws of the Catholic Church have varied through the years."
It is not necessary that all the laws of earlier Popes must be those of later Popes. Infallibility concerns faith and morals, not Church laws which fall into the category of Church discipline. Church laws adapted to particular times change with the times. Our grasp of a particular doctrine may change as we study it but this is not a change in the doctrine itself, but rather a growth in understanding of the teaching that was always there but may have been obscure to us.
Madrid, Pope Fiction; Drummey, Catholic Replies; Whitcomb, The Catholic Church has the Answer;Rumble and Carty, Radio Replies Vol 1; Defend the Faith, Vol 4 by Lumen Verum Apologetics.
* Please note that this text should be read in the context of the whole work and in recognition of the appropriate paragraphs of the Catechism of the Catholic Church highlighted in the index.