I have a fair bit of theological study and minored in Western History. First lets make sure we understand the limits of the period we are focusing on. That is the periods beginning c. 30 AD, the ministry of Jesus, up until one of these happened:
313, Emperor Constantine I legalized it with the Edict of Milan, certainly it would not fit under the adjectives you describe at this time.
325, Constantine I convened the First Council of Nicaea.
380, Emperor Theodosius I made it the state religion of Rome.
In any of the cases, by the time Constantine I was mixing it with state politics it was already changed from what it had been.
Now for what it had been. One important reminder is that many primary sources will be from Christians skewed in favor of their religion, and those who are against it and would aim to skew negatively towards it growth. And many secondary sources would come from later scholars, who just like the romanticization of the Roman Empire, would wish to create a narrative driven in favor of it.
With all of that in mind, we can get into the characteristics of it. One, state run persecution of it did not occur in a widespread manner until 246 under Decius. This was largely due to Decius demonizing them to help further the Imperial Cult, the religion of Rome at the time.
A unique aspect of Christianity is that it came from a religion that wasn't focused on proselytizing, Judaism, and yet focused on it heavily itself. This comes much from the 1st century Acts of the Apostles, also known as the book of Acts in the New Testament. Because of this focus on proselytizing, Christianity saw massive growth in a very short period, with 40% growth per decade for most of the 1st and 2nd century. [1] This early Christianity also lacked something very important. Because it spread so quickly it lacked both canon and structure for a considerable part of these 3 centuries. As new disciples of this faith rose up and spread it, more and more localized churches formed and with it imperfect scriptures. A few common characteristics did rise up though. One, the rejection of Judaism, often in violent manners, and the acceptance of Jesus' teachings. Calling it anti-establishment or radical is a bit of reach. The reason for this is that it didn't necessarily focus on state matters in the beginning, a teaching espoused by Jesus in his response on paying taxes to Rome. This lack of a focus on trying to affect state matters certainly helped in keeping the religious sects from being repressed by Rome. As the religion grew throughout the Mediterranean more and more texts appeared and were included or excluded by the heads of the churches. Once Constantine I held the council to determine what should and shouldn't be considered "Christianity" the religion saw a turn towards stability and unanimity. Where before texts could be interpreted or accepted by one church and not by another, and who is to say which is the real Christianity, now the state was involved and the Late Antiquity era of structure and togetherness of canon began.
I wrote my master's dissertation in ancient history on the episcopal subscription/attendance lists of the First Council of Nicaea (325), so this is definitely up my alley. There are so many ways to answer this prompt, but I'm going to look at it from the development of the church as an institution. By early Christianity I assume the Apostolic Age (death of Jesus to c. 100 CE) to the ante-Nicene period (100-325). By the 4th century, when Constantine had begun to patronize Christianity as his religion, and later on, the preferred religion of the Roman Empire, the Christian church already had a robust organizational structure, and this had been in development for almost 200 years.
The Didache, a handbook for the running of early churches dating to c. 100 CE, includes numerous regulations for early church ministers in which church leaders function as apostle, prophet, and teacher, referenced in 1 Corinthians 12:28 (NIV):
"And God has placed in the church first of all apostles, second prophets, third teachers, then miracles, then gifts of healing, of helping, of guidance, and of different kinds of tongues."
These three roles don't exactly exemplify the structure we would come to expect, but at the end of the Didache we see:
Elect for yourselves therefore bishops and deacons worthy of the Lord, gentle men, without avarice, true and proven; for they too minister to you the ministry of the prophets and teachers. Do not therefore despise them, for they are your honored ones along with the prophets and teachers. (compare with 1 Timothy 3:1-13)
So we see the election of regular officers alongside prophets and teachers. 'Deacon' comes from diakonos, meaning servant, whereas 'bishop,' episkopos, literally means an overseer. Writing at the turn of the 1st and 2nd centuries, Ignatius of Antioch urges each church to rally around the bishop and 'presbyters' (derived from presbys meaning elder) (Ign. Smyr. 8). It wasn't as cut and dry as it was later on, since a city could have multiple bishops around each respective Christian "household." Still, by the end of the 2nd century, Irenaeus and Tertullian expounded the concept of apostolic succession, that the apostles have passed down the truth to those whom each church was entrusted, the bishops, and at the beginning of each line of episcopal succession was an apostle from whom that church originated (Iren. Haer. 3.3.1-4; Tert. Praescr.). In the 3rd century during the height of Valerian's persecutions, Cyprian would go so far as to say that the episcopate was the sole judge of persons and groups in the church.
In many ways the church would parallel the imperial social order: clergy/laity vs plebs/curial class, Roman senators/equestrians/curiales etc. with bishops/presbyters/deacons and so on. On the whole we see the gradual increase in the authority of the bishop, and not only that, but a universal system of bishops united by their connection to the apostolic founders. Further parallels were drawn to the Aaron, the priestly brother of Moses, conflating the Jewish tradition as well. So by the mid-3rd century, the bishops functioned as both ecclesiastical governors and priests, and functioned as the conduit through which acceptable public worship was conducted.
Now we start to get into the politics. For as long as we have the church we have congregations that were the forerunners of church councils such as the congregation in Acts 1:15-26 which was conducted by Peter, Acts 13:1-3, and a formal delegation by Antiochene Christians to the 'apostles and presbyters' in Jerusalem (Acts 15:1-35). Of course, it might be a reconstruction by Luke, in which case it would tell us more about what church synods looked like in the late 1st century as opposed to the mid-century. Eusebius of Caesarea's Ecclesiastical History tells us that early heresies (late 2nd century) such as the New Prophecy (Montanism) were anathematized at councils:
When the faithful throughout Asia had met frequently and at many places in Asia for this purpose, and on examination of the new-fangled teachings had pronounced them profane and rejected the heresy, these persons were thus expelled from the church and shut off from its communion. (5.16.4-5)
Local assemblies could be visited by scholars or experts in consultation for local dogmatic issues such as heresy, with the object being consensus. Bishops in their growing roles as the embodiment of their local sees would correspond with other bishops on matters of worship, such as the date of Easter. We have firm evidence beginning c.250 onward for councils and procedures in the resolution for disputes, and now we see the developing structure of the church in action. Cyprian, faced with a schism of local clergy, gathers his own synod of local bishops, presbyters, and deacons and were able to establsih local guidelines (Cyp. Ep. 51.15.1).
As for anti-establishment, we see in the 3rd century the emperor Aurelian was asked to settle a theological dispute between Paul of Samosata and Domnus; Aurelian referred this matter to Pope Felix's discretion. Note how centrality of the bishops as representatives of the church to none other than the emperor himself.
So by the 4th century Constantine had found a church bound by this universal episcopal authority able to conduct its internal disputes from Rome all the way to Antioch - he himself organized a synod in Rome in 313, and another one in Arles in 314, for which we have the names of 33 bishops in the subscription lists attached to its canons. Of course, there was Nicaea. With imperial benefaction and encouragement, bishops were able to meet more frequently than ever before (Constantine granted bishops travelling to Nicaea full use of public roads and horses and subsidies for travel and lodging), and personalities could clash and ally, as ideas were shared and discussions held. Networks of correspondence facilitated the exchange of ideas or as some would say, heresies. Arius himself wrote to Eusebius of Nicomedia for support after his excommunication by Alexander of Alexandria, and Eusebius and Alexander as bishops would constitute the leaders of the opposing parties at Nicaea.
There's a lot more to say, especially on canons and the like, but I think this would be my initial answer, since I don't want it to go on for too long.
TL;DR Growth of the power of bishops under the authority of apostolic succession granted a cohesive hierarchy and structure to the church according to episcopal sees, and it was bishops who represented and embodied the churches of their cities in regional church politics in local synods and councils, as well as in the imperial arena.
Some core bib: Hall, Stuart George, Margaret Mary Mitchell, and Frances M. Young. "Institutions in the pre-Constantinian ecclesia." In The Cambridge History of Christianity. Volume 1. Origins to Constantine (2008): 415-433.
Graumann, Thomas. "The Conduct of Theology and 'Fathers' of the Church." In A Companion to Late Antiquity, edited by Philip Rosseau, 539-555. Oxford: Blackwell, 2009.
Edit: formatting and phrasing
By the time the first century CE rolled in, Judea, having been Hellenised through the conquest of Alexander the Great and the rule of the Seleucid Empire, the Hasmoneans, and Herodian client kings and formed into a semi-autonomous province of the Roman Empire, and the surrounding regions of the Levant and Mediterranean with whom it experienced cultural cross-pollination held a decently large contingent of φοβούμενοι τὸν Θεόν or "God-fearers," Gentile religious sympathizers to Hellenestic Judaism, a confluence of Second Temple Jewish practice and Greek thought. It is no accident that the first great theologian, apologist, and missionary of the Christian religion was Saulus (Paul), a Hellenized Jew and Roman citizen born in the city of Tarsus, Cilicia (modern day Turkey) and raised in Jerusalem.
It was in these communities of syncretized belief that Christianity first planted its roots and through them that its ideology spread across the furthest extents of the Roman Empire. For a long time, as the other commenter suggests, Christianity was relatively apolitical in terms of its relation to broader society, and, further, the proto-Orthodox period was incredibly diverse; though not fully distinct from Judaism in its first centuries, one moment of political difference was a failure to support the Jewish position of the Bar Kokhba Revolt. On the other hand, many Christian communities were intensely political, and even radical by some standards, but often only internally. Archeological evidence and external commentary support the position that the church was structured as reported in Acts as well as the Didache: the slave was maintained a slave societally but treated equally before the altar, the poor were cared for, and goods were shared in common.
Aside from the state persecution of the reign of Diocletian, Christians were largely assimilated into their communities, growing at a steady rate that hit a critical mass between 300 and 380 CE (~10% of the population to ~60%), at which point the Nicene formulation of Christianity was made the state religion of Rome by the cunctos populos (Edict of Thessalonica), to the exclusion of all other sects.
Sources for further reading:
The Rise of Christianity by Rodney Stark
All Things in Common by Roman A. Montero
The Cambridge History of Judaism, vol. 4 Chp. by Paul Trebilco; Davies et al.
Attitudes to Gentiles in Ancient Judaism and Early Christianity by Sim & MacLaren