According to the Catholic Church in her recent pronouncements, what is the order of truths to which the believer adheres?

Since this question has gone unanswered for some time, I will take a stab at it.

Levels of Magisterial Teaching is not the same thing as a hierarchy of truths.

Levels of Magisterial Teaching refers to the level of authority of the person or group giving the teaching and the circumstances under which it is given. So, for instance, the pope issuing a solemn ex cathedra document is accorded an infallible level of Magisterial Teaching. From elsewhere on the web:

The three levels of teaching are:

1) Dogma – infallible (Canon #750.1) to be believed with the assent of divine and Catholic faith.

2) Doctrine – infallible (Canon #750.2) requires the assent of ecclesial faith, to be "firmly embraced and held".

3) Doctrine – non-definitive (non-infallible) and requires intellectual assent ("loyal submission of the will and intellect", Vatican II, (Lumen Gentium 25), not an assent of faith.

The phrase a "hierarchy" of truths (quotes in the original) occurs in the second chapter of the document UNITATIS REDINTEGRATIO, otherwise known as the DECREE ON ECUMENISM, a Vatican II document, in the following paragraph:

Moreover, in ecumenical dialogue, Catholic theologians standing fast by the teaching of the Church and investigating the divine mysteries with the separated brethren must proceed with love for the truth, with charity, and with humility. When comparing doctrines with one another, they should remember that in Catholic doctrine there exists a "hierarchy" of truths, since they vary in their relation to the fundamental Christian faith. Thus the way will be opened by which through fraternal rivalry all will be stirred to a deeper understanding and a clearer presentation of the unfathomable riches of Christ.(34)

It's clear from the context that a "hierarchy" of truths is referring to the centrality of the content of a teaching to the Catholic faith. Doctrines related to the person of Jesus Christ are more central to the faith than doctrines related to church discipline, for example.

The Catechism of the Catholic Church starting at paragraph 80 makes clear that there is only one source of divine revelation, and we are obliged to adhere to everything that comes to us from the apostles without any order of importance or consideration of authority:

II. THE RELATIONSHIP BETWEEN TRADITION AND SACRED SCRIPTURE

One common source. . .

80 "Sacred Tradition and Sacred Scripture, then, are bound closely together, and communicate one with the other. For both of them, flowing out from the same divine well-spring, come together in some fashion to form one thing, and move towards the same goal."40 Each of them makes present and fruitful in the Church the mystery of Christ, who promised to remain with his own "always, to the close of the age".41

. . . two distinct modes of transmission

81 "Sacred Scripture is the speech of God as it is put down in writing under the breath of the Holy Spirit."42

"And [Holy] Tradition transmits in its entirety the Word of God which has been entrusted to the apostles by Christ the Lord and the Holy Spirit. It transmits it to the successors of the apostles so that, enlightened by the Spirit of truth, they may faithfully preserve, expound and spread it abroad by their preaching."43

82 As a result the Church, to whom the transmission and interpretation of Revelation is entrusted, "does not derive her certainty about all revealed truths from the holy Scriptures alone. Both Scripture and Tradition must be accepted and honored with equal sentiments of devotion and reverence."44

Apostolic Tradition and ecclesial traditions

83 The Tradition here in question comes from the apostles and hands on what they received from Jesus' teaching and example and what they learned from the Holy Spirit. The first generation of Christians did not yet have a written New Testament, and the New Testament itself demonstrates the process of living Tradition.

Tradition is to be distinguished from the various theological, disciplinary, liturgical or devotional traditions, born in the local churches over time. These are the particular forms, adapted to different places and times, in which the great Tradition is expressed. In the light of Tradition, these traditions can be retained, modified or even abandoned under the guidance of the Church's Magisterium.