First off, it is a tradition that we call priests Father and there is a reason for it. Paul talks about how he is a spiritual father to the early Christians, and so too the priests of today follow his lead and that of the other Apostles in fathering and shepherding the Church.
I can only speak for what goes on in the seminaries in the United States with regard to degrees conferred and the way the studies go. Other countries will do things differently, and perhaps others can speak to that. In the US, one must study for either 8 years or 6, depending upon his educational background.
With no college degree, it is 8 years–4 of college seminary, culminating in a Bachelor of Arts degree, usually with a concentration in Philosophy (although others are possible–History and Humanities, and others, depending on the seminary. Regardless of concentration, college seminarians are required to take a certain number of Philosophy credits in order to advance to Theological studies). With a college degree, one enters a program called Pre-Theology, which is an abbreviated Philosophy program. He still takes the same number of required Philosophy credits, but doesn’t take the core-level general education classes (in Math, Science, English, etc) required of a seminarian pursuing a BA, since he would already have done this as an undergraduate. Most seminaries offer a Bachelor of Philosophy degree to Pre-Theologians completing the program; some simply offer a certificate verifying that the seminarian has completed all his Philosophy credits.
After either of these follows 4 years of graduate-level Theology, culminating in a minimum of a Master of Divinity, the basic credential for ministry in the US. In most seminaries it is possible (and optional) as well to earn a Master of Arts in Theology, at the discretion of the seminarian and with the advice and consent of his Diocese and formation faculty at the seminary. In this instance, a concentration is offered. At my own seminary, the MA can be earned in either Scripture, Moral Theology, Dogmatic Theology, or Evangelization. Some seminaries offer an additional (and also optional) Bachelor of Sacred Theology, in conjunction with one of the Pontifical Universities in Rome.
I have already somewhat answered the question on specialization; I might add as well that if the Bishop so desires, he may send his priests on for further studies beyond seminary. This is most typically done in Canon Law, so there will be trained canonists in the diocese, but some dioceses send priests for other disciplines as well, and these priests get assigned in teaching jobs or as theologians.
I hope I have answered your questions clearly. May God bless you.