Jungmann, Josef A., S.J. "Chapter Seven: Baptism and Preparation for Baptism." The Early Liturgy to the Time of Gregory the Great. (translated by Francis A. Brunner, C.S.S. R., Notre Dame, Indiana: University of Notre Dame Press, 1959, pp. 74-86.
Jungmann sees Christianity as growing very strong in the third century. "In 250, it is stated that the Roman community regularly supported 1500 poor; hence the faithful altogether must have numbered many tends of thousands. At the time of Diocletian's persecution, there were as many as forty basilicas in the city" (Jungmann 1959, 74). Jungmann does not cite a source for this information. However, on the next page he does reference Harnack, Die Mission und Ausbreitung des Christentums in den ersten drei Jahrhunderten (1902). Jungmann does suggest, though, that judging from Hippolytus' Church Order, the stringent conditions to be admitted to catechesis may have been related to the growth. The standards may have shown potential adherents they needed to be serious (Jungmann 1959, 75). Jungmann describes Hippolytus' intervie wprocess in brief.
Catechesis was also challenging. It typically lasted three years, focused on moral trial and an entry into the corporate life of the Church (Jungmann 1959, 77). The candidate for baptism could then be examined, then enter into theological studies in the period leading up to Easter (Jungmann 1959, 78). Then, in the time shortly before baptism, the candidate would participate in special prayers, fasting, and exorcisms. Baptism would take place in a night vigil. This would include the adult catechumens and children who may not have been old enough to answer for themselves (Jungmann 1959, 79).
The baptism of infants was common. Jungmann describes a controversy in northern Africa about the year 250, in which a bishop Fidus taught that parents should wait until the eighth day after birth, to parallel circumcision. A synod of 66 bishops "unanimously decided that no one should wait till the eighth day, but that the babies of Christian parents should be baptized immediately" (Jungmann 1959, 80).
The baptism included a renunciation of Satan, a profession of faith and a triple immersion (Jungmann 1959, 81). Jungmann does note that infusion rather than immersion is allowed in the Didache (Jungmann 1959, 82). Jungmann describes the steps of baptism in some detail (Jungmann 1959, 83-84). After baptism, the neophytes would enter into a period of catechetical instruction, normally for about a week (Jungmann 1959, 85).