Religious doctrines and practices rarely appear fully-formed - they are generally the product of lengthy reflection, refinement and development.
The Holy Trinity
This is a classic example of the progressive development of doctrine.
Neither Jesus nor St Paul, the great theologian of early Christianity, taught the doctrine of the Holy Trinity. It would probably have been too much of a leap from their native Judaism. Paul consistently distinguishes between God the Father and Jesus the Lord, who intercedes with God for mankind but is not himself divine. Only a few verses in the New Testament teach the doctrine of the deity of Jesus (for one famous example, see the beginning of St John’s Gospel - ‘In the beginning was the Word [i.e. Christ], and the Word was with God, and the Word was God’). Likewise, the Holy Spirit tends not to appear in the New Testament as a personal entity, but rather as the impersonal action of God on the world.
Up until around 150 AD, Christian writers generally seem to follow St Paul in distinguishing between Christ and God, though their Christology - their view of the nature of Christ - is not always clear. One important exception is Ignatius of Antioch (c.110 AD), who unequivocally referred to Jesus as God. After the middle of the century, the deity of Christ becomes a commonplace of Christian theology. Trinitarian ideas also start to appear in Christian writings, though these too had been prefigured by Ignatius. The first clear presentation of Trinitarian doctrine which I know of is found in a work known as 2 Clement which dates from c.150. Less clear traces of the doctrine are also found in the works of Aristides, Justin Martyr and Diognetus, which were composed in the 140s and 150s.
Trinitarian doctrine was quite firmly established in the third century, and finally triumphed after the Council of Nicaea (325).
The first leaders of the Christian movement were the twelve apostles, headed by St Peter. These were quickly joined by the new convert St Paul, who believed that God had specially commissioned him as an apostle, and by Jesus’ brother St James, to whom Jesus was believed to have appeared after his resurrection.
St Paul’s letters, written in the 50s and early 60s AD, refer to a number of different personalities within the communities with which Paul dealt, including apostles, prophets, teachers, miracle-workers, healers and administrators (see esp. 1.Cor. 12.28). The structures of authority still seem to be fairly vague and fluid. On the other hand, we can see the first traces of the classical hierarchy of bishops (literally, "overseers"), priests (literally, "elders") and deacons: at Phil. 1.1, Paul refers to overseers and deacons at Philippi, and Rom. 16.1 attests to the existence of female office-holders called deaconesses.
In the 70s and 80s, elders start to be mentioned for the first time, notably in Acts (11.30, 14.23, 15.2, 20.17, 28), where they are presented as running the church and are also referred to as overseers. A roughly contemporary letter known as 1 Clement also makes reference to elders and overseers as the governors of autonomous Christian communities, and seems to regard the two offices as identical; he also knows of deacons. Prophets are mentioned at Mt. 7.15 and 23.34.
In the final years of the century, James 5.14 and 1 Pet 5.1 refer to elders as being in charge of contemporary Christian communities. Elders also appear in 1 Tim. 5, though the preceding chapters of the same epistle seem to present local churches as being governed by a single overseer (they also mention deacons). The move towards a monarchic episcopacy had begun.
It had not, however, ended - not until c.150 do singular overseers distinct from the elders emerge as the undisputed leaders of the church communities, and texts such as the letters of Ignatius of Antioch (c.110) which refer to singular overseers are still the exception rather than the rule. In the first half of the second century, it seems that most communities continued to be run by ‘colleges’ of elders who also called themselves overseers, and significant traces of the older arrangement appear to have lingered on for some decades afterwards. Deacons also continue to be mentioned in second-century literature. In the third century, references to the tripartite hierarchy of bishops/overseers, priests/elders and deacons become commonplace.
Charismatic leader-figures such as prophets seem to disappear quite early in the second century, though a reference is made to them in the Didache (c.140). By the end of the second century, writers such as Irenaeus and Tertullian were formulating the classical doctrine of the apostolic succession, which left no room for leadership outside the institutional structures of the Church. The doctrine recurs repeatedly in third-century writings.
The Roman primacy
The first possible reference to the primacy of the church of Rome comes in the writings of Ignatius of Antioch (c.110). Ignatius refers to the church of Rome as ‘having the leading place’ among the Christian communities. An earlier letter (known as 1 Clement), dating from the second half of the first century, was sent by the church at Rome to the church at Corinth. This expresses the anger of the Romans at recent misdeeds committed by the Corinthians, though it doesn't really give the impression that the church of Rome is seeking to exercise a prerogative of leadership that belongs to it by right.
The idea of the primacy of the church of Rome appears in the writings of Irenaeus (180+), though Tertullian, writing around the year 200, disallowed that the church of Rome had a special status. The notion that the Roman church and its leader possessed an overarching authority began to gather strength in the third century.
Rome was not typically celebrated in early Christian literature as the see of Peter - rather, the apostles Peter and Paul were both associated with the city, and there wasn't the sense that the bishop of the city was the successor of the chief apostle.
Proto-Protestant ideas about salvation were current in the late first and early second century. Drawing upon the writings of St Paul, a number of writers seem to teach something approximating to sola fide (the doctrine of salvation by faith alone), as well as positing a dichotomy between the saved elect and the unsaved multitudes. All this is more than a little reminiscent of later Protestant theology, and of Calvinism in particular (though, interestingly, St Paul himself seems not to have taught sola fide).
This proto-Protestantism seems to disappear around the middle of the century, and by the 190s a writer like Clement of Alexandria was able to explicitly disavow the doctrine of salvation by faith alone.
The resurrection of believers
The resurrection of individual believers - as opposed to the resurrection of Christ - was a very important part of second- and early third-century Christian doctrine. Christians repeatedly looked forward to the day when history would come to an end and the righteous would be raised up in glorified bodies just as Jesus had been raised up by his father. Though the resurrection of believers has always remained part of Christian doctrine, it suddenly seems to fade into the background around the middle of the third century.
The first reference to the practice of confession and absolution comes in the writings of Tertullian (c.200). Tertullian was not happy with the idea that Christians would need to go to confession after being baptised, but he allowed that one single post-baptismal confession was permissible.
Confession is mentioned with some frequency in the third century, though Tertullian’s belief that it should not be repeated more than once remained very influential. It is generally clear that absolution must be granted by a priest.
Confirmation is a ceremony of initiaiton following baptism in which the new Christian receives the Holy Spirit. It is first mentioned in the Acts of the Apostles, where receiving the Holy Spirit seems to be a synonym for receiving the power to perform miracles such as healings. Thereafter, the record seems to be silent until the third century, when the sacrament is mentioned in the works of the writers Cornelius (251-253) and Cyprian (240s/50s).