Hi there.
I'm a seminarian, so I can answer your questions.
(To an extent, each diocese and seminary is different)
To become a priest, you have to apply (Diocese or order)
For me, it took about six months of Interviews, paperwork, and exams (Both physical and psychological)
After being accepted, a seminarian is required to go to a college seminary, that is, if he does not already have a degree. This is the normal college of four years. And yes,you major in philosophy. If the seminarian already has a degree, he will be required to get another degree in Philosophy. (In my diocese, this takes two years.)
After College-Seminary, the seminarians move on to Major Seminary, where they study Theology for four years, focusing on practicum for the last two. (IE, learning how to be pastoral, homiletics, offer the sacraments, et c.)
Additional, in some dioceses, there is an additional year of working in a parish. In Florida it is wedged in between II and III Theology and called "Pastoral Year". This is more or less an internship.
At the end of III Theology, the seminarian is ordained a deacon. And at the end of IV theology, he is a priest.
All-in-all, it takes between 6 and 9 years to become a priest.