According to the Catholic Church, are there levels of spiritual enlightenment in Christianity? In Judaism there is the Kabbalah. In other religions there are ways to spiritual enlightenment.
Every religion has its own formula.

From everyone who has been given much, much will be demanded; and from the one who has been entrusted with much, much more will be asked. (Luke 12:48)
I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ. (Galatians 1:11)
was caught up to paradise and heard inexpressible things, things that no one is permitted to tell. (2 Corinthians 12:4)

So are there levels of enlightenment in Christianity? If so, how do we get there? What is the Catholic Church's doctrine on this question?
I believe you are looking for the three ways or states of the spiritual life.


  Our process of conversion to Christ is a journey that takes place over the course of a lifetime. You can probably look back to times in your life where you can remember making great progress on your spiritual journey, and other times when you have felt like you were moving backwards.
  
  The Fathers, theologians, and saints of the Church discuss three stages, or states, of perfection in the spiritual life: the purgative way, the illuminative way, and the unitive way. Each “way” or state represents an advancement toward sanctity—perfection—and total union with God.
  
  Here is a quick synopsis of the ways:
  
  Purgative Way:
  
  This is the state of beginners. In this stage it is often difficult to overcome daily temptations, and practicing the virtues can require an inner battle because of attachment to venial sin. Though the desire for perfection and progress is there, the beginner falls frequently. If a beginner concerns himself seriously with repenting of his sins and has an actionable desire to stop offending God, he may eventually move to the Illuminative Way.
  
  Illuminative Way:
  
  This is the intermediary stage between purification and total union with God. In this stage, there is enlightenment in the ways of God and a clear understanding of his will in one’s life. There are now only occasional “slips” into sin.
  
  Unitive Way:
  
  This is the final stage of Christian perfection. A soul in the unitive state has a constant awareness of God’s presence and habitually conforms to God’s will. In this stage, the soul loves God and others without limit. - The Three Ways or States of the Spiritual Life


Many Catholic authors have written extensively on the subject of these three stages of the spiritual life including Fr. Reginald Garrigou-Lagrange, O.P. (in The Three Ages of the Spiritual Life: Prelude of Eternal Life) and St. John of the Cross (in The Dark Night of the Soul).

cf. St. Thomas Aquinas's Summa Theologica II-II q. 24 a. 9 c.:


  The spiritual increase of charity may be considered in respect of a certain likeness to the growth of the human body. For although this latter growth may be divided into many parts, yet it has certain fixed divisions according to those particular actions or pursuits to which man is brought by this same growth. Thus we speak of a man being an infant until he has the use of reason, after which we distinguish another state of man wherein he begins to speak and to use his reason, while there is again a third state, that of puberty when he begins to acquire the power of generation, and so on until he arrives at perfection.
  
  In like manner the divers degrees of charity are distinguished according to the different pursuits to which man is brought by the increase of charity. For at first [purgative way:] it is incumbent on man to occupy himself chiefly with avoiding sin and resisting his concupiscences, which move him in opposition to charity: this concerns beginners, in whom charity has to be fed or fostered lest it be destroyed: in the second place [illuminative way:] man's chief pursuit is to aim at progress in good, and this is the pursuit of the proficient, whose chief aim is to strengthen their charity by adding to it: while man's third pursuit [unitive way:] is to aim chiefly at union with and enjoyment of God: this belongs to the perfect who "desire to be dissolved and to be with Christ."


The Catholic Encyclopedia has an excellent article on this subject.

It could be noted that the unitive way is the stage which some souls receive the gifts of contemplation and union with God.


  It is in this state that the gift of contemplation is imparted to the soul, though this is not always the case; because many souls who are perfect in the unitive way never receive in this life the gift of contemplation and there have been numerous saints who were not mystics or contemplatives and who nevertheless excelled in the practice of heroic virtue. Souls, however, who have attained to the unitive state have consolations of a purer and higher order than others, and are more often favored by extraordinary graces; and sometimes with the extraordinary phenomena of the mystical state such as ecstasies, raptures, and what is known as the prayer of union. -  Catholic Encyclopedia.