The Church's Theological Notes or Qualifications

Listed below are the so-called theological notes and their associated censures from the table in Sixtus Cartechini, S.J.'s 1951 work De Valore Notarum Theologicarum (On the Value of the Theological Notes), which confessors have used when dealing with erudite penitents; it's also available in Italian translation. The theological notes are a way of classifying the proximity of a theological proposition to Divine Revelation. For a good history of the development of these notes, see The development of the theological censures after the Council of Trent: (1563-1709) by John Cahill, O.P.

1. Theological note: Dogma.
   Equivalent terms: Dogma of faith; de fide, de fide Catholica; de fide divina et Catholica.
   Explanation: A truth proposed by the Church as revealed by God.
   Examples: The Immaculate Conception; all the contents of the Athanasian Creed.
   Censure attached to contradictory proposition: Heresy
   Effects of denial: Mortal sin committed directly against the virtue of faith, and, if the heresy is outwardly professed, excommunication is automatically incurred and membership of the Church forfeited.
   Remarks: A dogma can be proposed either by a solemn definition of pope or council, or by the Ordinary Magisterium, as in the case of the Athanasian Creed, to which the church has manifested her solemn commitment by its long-standing liturgical and practical use and commendation.
2. Theological Note: Doctrine of ecclesiastical faith
   Equivalent term: De fide ecclesiastica definita
   Explanation: A truth not directly revealed by God but closely connected with Divine revelation and infallibly proposed by the Magisterium.
   Example: The lawfulness of communion under one kind.
   Censure attached to contradictory proposition: Heresy against ecclesiastical faith.
   Effects of denial: Mortal sin directly against faith, and, if publicly professed, automatic excommunication and forfeiture of membership of Church.
   Remarks: It is a dogma that the Church's infallibility extends to truths in this sphere, so one who denies them denies implicitly a dogma or Divine faith.
3. Theological Note: Truth of Divine faith.
   Equivalent term: De fide divina.
   Explanation: A truth revealed by God but not certainly proposed as such by the Church.
   Example: Christ claimed from the beginning of His public life to be the Messias.
   Censure attached to contradictory proposition: Error (in faith).
   Effects of denial: Mortal sin directly against faith, but no loss of Church membership. May incur a canonical penalty.
4. Theological Note: Proximate to faith.
   Explanation: A doctrine all but unanimously held as revealed by God.
   Example: Christ possessed the Beatific Vision throughout his life on earth.
   Censure attached to contradictory proposition: Proximate to error.
   Effects of denial: Mortal sin indirectly against faith.
5. Theological Note: Theologically certain.
   Equivalent term: Dogmatic fact; theological conclusion.
   Explanation: A truth logically following from one proposition which is Divinely revealed and another which is historically certain.
   Example: Legitimacy of Pope Pius XI.
   Censure attached to contradictory proposition: Error (in theology).
   Effects of denial: Mortal sin against faith.
6. Theological Note: Catholic doctrine.
   Equivalent term: Catholic teaching.
   Explanation: A truth authentically taught by the Ordinary Magisterium but not as revealed or intimately connected with revelation.
   Example: Invalidity of Anglican Orders; validity of Baptism conferred by heretic or Jews.
   Censure attached to contradictory proposition: Temerarious.
   Effects of denial: Mortal sin indirectly against faith.
   Remarks: The expression Catholic doctrine is sometimes applied to truths of a higher order also, but never of a lower one. In some cases the appropriate censure may be graver than "temerarious".
7. Theological Note: Certain.
   Equivalent term: Common; theologically certain.
   Explanation: A truth unanimously held by all schools of theologians which is derived from revealed truth, but by more than one step of reasoning.
   Example: The true and strict causality of the sacraments.
   Censure attached to contradictory proposition: Temerarious.
   Effects of denial: Usually, mortal sin of temerity.
   Remarks: Proportionately grave reason can sometimes justify an individual who has carefully studied the evidence in dissenting from such a proposition; since it is not completely impossible for all the theological schools to err on such a matter, although it would be highly unusual and contrary to an extremely weighty presumption.
8. Theological Note: Safe.
   Explanation: Affirmed in doctrinal decrees of Roman Congregations.
   Example: That Christ will not reign visibly on earth for a thousand years after Antichrist.
   Censure attached to contradictory proposition: Unsafe/temerarious.
   Effects of denial: Mortal sin of disobedience and perhaps imprudence.
   Remarks: Exterior assent is absolutely required and interior assent is normally required, since, though not infallible, the Congregations possess true doctrinal authority and the protective guidance of the Holy Ghost.
9. Theological Note: Very common/commoner.
   Explanation: The most solidly founded or best attested theological opinion on a disputed subject.
   Example: Antichrist will be of the tribe of Dan.
   Censure attached to contradictory proposition: None.
   Effects of denial: None.
   Remarks: Very common or commoner opinions can be mistaken and there is no obligation to follow them though prudence inclines us to favour them as a general policy. It should be noted that an opinion which is "very common" is less well established than one which is "common" which implies moral unanimity of theological schools.
10. Theological Note: Probable.
    Explanation: A theological opinion which is well founded either on the grounds of its intrinsic coherence or the extrinsic weight of authority favouring it.
    Example: Judas received Holy Communion at the Last Supper. Judas did not receive Holy Communion at the Last Supper.
    Censure attached to contradictory proposition: None.
    Effects of denial: None.
    Remarks: The better founded of two conflicting opinions is referred to as more probable; but Catholics are free to prefer some other opinion for any good reason.

Compared with the "Wesleyan Quadrilateral"

Thus, to answer the question:

It seems it could (very loosely) be ordered this way, in terms of weight:

1. Tradition,
2. Scripture,
   Scripture would come after Tradition because a truth of divine faith (de fide divina) is ranked #3—after a Doctrine of ecclesiastical faith (de fide ecclesiastica), #2, or a defined dogma (dogma de fide), #1; all truths in Holy Scripture that have not been explicitly been defined as dogmas are truths of divine faith (de fide divina).
   
   However, it's not quite correct to place Scripture as less weight than Tradition because Scripture is a part of Tradition.
3. human reason.
   This ranks after Tradition and Scripture because reason is subject to faith, pace the rationalists. See: the First Vatican Council's section in Dei Filius on Faith & Reason.
   
   The rest of the theological notes could loosely be considered in the category of "reason," but this is not quite correct because theological conclusions (note #5) can include one premise from natural reason and another from Revelation. Here's an example, where the major premise comes from Revelation and the minor premise from human reason:
   
   
   
   Also, human reason would come after Tradition or Scripture because sacred doctrine (sacra doctrina) is the supreme science, being the noblest and most certain.
   
   Ranking human reason #3 is also problematic because certain truths (the præambula fidei or "preambles of faith") are required first before Tradition, Holy Scripture, and dogma can even be understood in the first place; so, in this sense, certain perennial truths of human reason (which the philosophers have discovered, e.g., that truth exists, humans can attain truth, humans have immortal souls, etc.) would rank first. These truths are listed in De Valore Notarum Theologicarum ch. 6 on theologically certain propositions.

If "experience" means "private revelation," it would not appear here. Private revelations do not of themselves have Magisterial authority, although the Magisterium can declare them as free of error. If "experience" does not mean "private revelation" but something along the lines of "that which helps us reason," it would be ranked #3 alongside human reason.

From what has been said above, it appears that the Catholic understanding cannot neatly fit into a "Wesleyan Quadrilateral."

The loci theologici

The loci theologici are what you seem to mean by "different sources of information" which theologians use "in coming to doctrinal/theological conclusions." Melchior Cano (1509-1560) treated them his De Locis Theologicis. They are:

• the authority of
  1. Holy Scripture,
  2. Catholic tradition,
  3. general councils,
  4. the Roman Church,
  5. the Fathers,
  6. the Schoolmen;
  7. natural reason,
  8. philosophers and doctors in civil law,
  9. history.

Cano said the last three are useful but not necessary, and he thought there could be more.