Which doctrines are Catholics required to accept? I only recently figured out that some Catholic doctrines are required for a Catholic to accept, while the denial of others doesn't prevent being a Catholic.

Examples of both cases:


Mark Trapp, in his answer to List of papal teachings considered infallible, quotes two instances of ex cathedra, both of which contain a judgment of those who don't accept the doctrine. The declaration of Immaculate Conception, for example, ends with


  Hence, if anyone shall dare—which God forbid!—to think otherwise than as has been defined by us, let him know and understand that he is condemned by his own judgment; that he has suffered shipwreck in the faith; that he has separated from the unity of the Church...

The doctrine of papal infallibility was not a strictly required one in 1860, as Keenan's Catechism used in British Catholic schools included the following: (as quoted in Wikipedia)


  (Q.) Must not Catholics believe the Pope himself to be infallible?
  (A.) This is a Protestant invention: it is no article of the Catholic faith: no decision of his can oblige under pain of heresy, unless it be received and enforced by the teaching body, that is by the bishops of the Church.



What are the distinctions between these two kinds of doctrines? Is there a list of required doctrines somewhere?
The degree to which Roman Catholics need to accept doctrines as required is directly related to how certain they are known to be true. There is a range of levels of certitude, from immediately revealed truths (fides divina) and the infallible teachings of the Church (fides ecclesiastica) as the most certain to things commonly believed to be true by most theologians (sententia communis) down to things tolerated by the Church but not really well founded (opinio tolerata). 

To wit, there are three classes of dogma:


truths formally and explicitly revealed
truths revealed formally but not explicitly revealed
truths known through their connection to other truths, but not revealed


The first two are what generally constitutes necessary dogma: things every person needs to believe in order to be considered in communion with the Church. These are well-known and binding: things like the Apostle's Creed, the Nicene Creed, the primacy of the Pope and the Apostolic succession, and so forth.

The third category is where there is some leeway in what is necessary to believe. These are open for debate amongst theologians, and include things like whether the existence of a proof for God exists, whether we can have knowledge of Heaven, etc.

The part which you quote from the pronouncements made ex cathedra is actually procedural and is meant to classify the teaching as formally revealed; the First Vatical Council laid out conditions that need to be met for a teaching to be considered made ex cathedra:


  Therefore, faithfully adhering to the tradition received from the beginning of the Christian faith, to the glory of God our savior, for the exaltation of the Catholic religion and for the salvation of the Christian people, with the approval of the Sacred Council, we teach and define as a divinely revealed dogma that when the Roman Pontiff speaks EX CATHEDRA, that is, when, in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, he defines a doctrine concerning faith or morals to be held by the whole Church, he possesses, by the divine assistance promised to him in blessed Peter, that infallibility which the divine Redeemer willed his Church to enjoy in defining doctrine concerning faith or morals. Therefore, such definitions of the Roman Pontiff are of themselves, and not by the consent of the Church, irreformable.
  
  So then, should anyone, which God forbid, have the temerity to reject this definition of ours: let him be anathema.


That is, the formula is:


The Holy Father speaks in his official capacity


  by the authority of Jesus Christ our Lord, of the Blessed Apostles Peter and Paul, and by our own

He speaks definitively   


  we declare, pronounce, and define

Anyone who doesn't believe what was taught has left the Church


  Hence, if anyone shall dare—which God forbid!—to think otherwise than as has been defined by us, let him know and understand that he is condemned by his own judgment; that he has suffered shipwreck in the faith; that he has separated from the unity of the Church



And so to reconcile with the question and answer pair in the Keenan's Catechism, it's important to note that the pope himself is not infallible: he only has the ability to speak infallibly, and only when he speaks ex cathedra. This doesn't happen all the time: it's a very rare occurrence used only in very specific situations (to date, only about the Marian dogmas mentioned in my answer). 

Instead, most of the dogma Roman Catholics believe comes directly from the Magisterial teachings of the Church. The pope can speak about those infallible teachings, but he speaks about them in a fallible manner (i.e. he can misunderstand or misinterpret them, as he's only human).

As for lists of necessary dogmata, the Catechism of the Catholic Church is the formal collection of what the Church believes. Peter Turner in his answer to the question about the list of teachings considered infallible mentioned another great resource, Fundamentals of Catholic Dogma, which summarizes nearly every dogmatic teaching.