Presbyters, Bishops, and Deacons in the New Testament
The Acts of the Apostles and the Epistles make frequent mention of presbyteroi - literally "elders", sometimes also translated as "presbyters". These were often appointed directly by Apostles, as was, for example, the case in Lystra, Iconium, and Antioch:
Confirming the souls of the disciples, and exhorting them to continue in the faith, and that we must through much tribulation enter into the kingdom of God. And when they had ordained them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed.
In other cases, they were appointed by those assigned to do so by the Apostles, as was the case with Timothy and Titus:
1 Timothy 4:14 (KJV)
Neglect not the gift that is in thee, which was given thee by prophecy, with the laying on of the hands of the presbytery.
For this cause left I thee in Crete, that thou shouldest set in order the things that are wanting, and ordain elders in every city, as I had appointed thee:
We know from James that there could often be multiple presbyters in one city:
Is any sick among you? let him call for the elders of the church; and let them pray over him, anointing him with oil in the name of the Lord
The Greek word presbyter was carried over into Latin and eventually contracted to form the Old English word "preost" and German "Priester", leading eventually to the Modern English word "priest".
Certain presbyters (or elders, if one prefers that term) were appointed to oversee others, sometimes distinguished with the special title of episkopos, which was adopted in Old English as "biscop" or "biscoep" and has since found a place in English as the word "bishop". The word "bishop" is sometimes used to translate episkopos, but often the term "overseer" is preferred. When Paul instructed the presbyters of the Church from Ephesus (Acts 20:17,28), he said:
Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers [episkopoi], to feed the church of God, which he hath purchased with his own blood.
A third Church office discussed in the New Testament is that of diakonos, usually translated as "deacons". These were assigned to assist the Apostles and the faithful in general. The role of deacons in the time of the New Testament is described in Michael Pomazanksi's Orthodox Dogmatic Theology (3rd ed.):
From the pastoral epistles it is apparent that the deacons were appointed by bishops (I Tim. 3: 8– 13). According to the book of Acts, for the ministry of deacon there were chosen people “filled with the Holy Spirit and wisdom” (Acts 6: 3). They took part in preaching, as did St. Stephen, who sealed his preaching of Christ with his martyr’s blood; and like St. Philip, who performed the Baptism of the eunuch (Acts 8: 5 and 38). In the Epistle to the Philippians, the Apostle Paul sends greetings to “the bishops and deacons” (1: 1), as bearers of the Grace-given hierarchical ministry, helpers of the bishops.1
In the Greek-speaking Churches of today, these three offices are known by precisely the same Greek terms found in the New Testament: episkopos, presbyteros, and diakonos. In English, as discussed above, they have been largely replaced by "bishop", "priest", and "deacon" (although the term "presbyter" is still often used in the Orthodox Church).
Developments in the 1st and 2nd Centuries
As is clear from the above, although the three offices of Presbyter, Bishop, and Deacon are mentioned in some form in the New Testament, the hierarchy was not nearly as structured as it appears today in those Christian confessions that attempt to maintain it. As you rightly point out, one - in the absence of any other context - could extrapolate all sorts of structures from what we read about in the New Testament, and perhaps make arguments for or against them based on various doctrinal positions.
As a practical matter, though, the fairly informal structure outlined in Acts and the Epistles would have become (and did become) unmanageable as Christianity spread both in geography and population. We see even by the sixth chapter in Acts that the Apostles were already struggling to practically minister to the relatively small flock at the time.
As a result, the offices of bishop and presbyter came into sharper focus sometime during the end of the Apostolic era and at the very beginning of the post-Apostolic era, when a bishop became clearly responsible for a particular city, overseeing a number of presbyters, with deacons assisting overall. Metropolitan Kallistos Ware writes:
Within an astonishingly short time small Christian communities had sprung up in all the main centres of the Roman Empire and even in places beyond the Roman frontiers.
The Empire through which these first Christian missionaries travelled was, particularly in its eastern part, an empire of cities. This determined the administrative structure of the primitive Church. The basic unit was the community in each city, governed by its own bishop; to assist the bishop there were presbyters or priests, and deacons. The surrounding countryside depended on the Church of the city. This pattern, with the threefold ministry of bishops, priests, and deacons, was already established in some places by the end of the first century. We can see it in the seven short letters which St Ignatius, Bishop of Antioch, wrote about the year 107 as he travelled to Rome to be martyred. Ignatius laid emphasis upon two things in particular, the bishop and the Eucharist; he saw the Church as both hierarchical and sacramental. ‘The bishop in each Church,’ he wrote, ‘presides in place of God.’ ‘Let no one do any of the things which concern the Church without the bishop… Wherever the bishop appears, there let the people be, just as wherever Jesus Christ is, there is the Catholic Church.’
As the Church grew even further, it became necessary to put some sort of structure in place to further organize the bishops. In the east, the bishops of very large cities came to be known as "Metropolitans". By the 4th century, Christendom came to be divided into five administrative "mega-centers" called "Sees": Rome (the old capitol of Rome), Constantinople (the new capitol of Rome), Alexandria, Antioch, and Jerusalem. Bishops appointed over these very large regions were known as "patriarchs" (patriarchoi), with the Patriarchs of Rome (and Alexandria) sometimes referred to with the term of affection, pappas - the origin of the word "Pope".
2. The Orthodox Church, p.12