To what extent do the ideals of the Renaissance anticipate those we associate with modernity? To what extent are they more faithful to either the Christian or the classical past?

The Renaissance was quite modern in the sense that it focused on learning. People looked for rational reasons for the natural phenomena in their world. With the advent of the printing press, there was a thirst for books, and booksellers became wealthy in major cities. They were also able to ship their goods throughout Europe as people traveled more. People looked for their own interpretations of religion, away from the teachings of the Catholic Church and based on their own interpretations of the Bible. People wrote "how-to" books about manners, leadership, and military science. Today, the "how-to" book remains quite popular.
The Renaissance still had roots in the Christian and Classical past in that it drew heavily from the ancient Greek and Roman authors who were shunned by the Catholic Church because they were not Christian. People even adopted names similar to these authors when they wanted to create a pseudonym. The whole idea of the Renaissance was to recapture the "glory days" of Western Europe when it was influenced by Greek and Roman thought. The leaders of the Reformation sought to re-create the Church as it was understood by Christ and the apostles before it became influenced by the Catholic Church. In this respect, the Renaissance was a classical movement as well as a modern one.