What was it like to be a christian in first century?

This is a somewhat difficult question to answer, in part because referring to followers of Jesus in the 1st century as Christians may be anachronistic in many cases; scholars tend to describe what might be called "proto-Christians" as "the Jesus movement" to distinguish it from the later, more institutionally and theologically developed Christian church. Scholarship has come to see the experience of proto-Christians and followers of Jesus in this period as incredibly diverse and varied from region to region, city to city. First of all, doctrine was even less firmly established at this time than even during the great debates on christology in the 3rd and 4th centuries, and there are plenty of clues in the New Testament that point to this fact. Paul, in his first epistle to the Thessalonians, writes to allay the concerns of proto-Christians who were disturbed by the death of some members of their community, the implication being that they did not expect anyone of them to die before Jesus' return. In Galatians, he addresses a dispute between Gentile converts and Jewish members over whether the former must follow the Mosaic Law; Paul asserts (contrary to Peter and the other Twelve) adamantly that they do not, but this is not what the author of the Gospel of Matthew, writing about 30 years later says, when he quotes Jesus in 5:17 as saying "not one dot, not one iota of the Law will pass away," showing that the dispute over the Mosaic Law was still ongoing decades after Paul wrote. These are just a few of the myriad examples of what it was like to be a proto-Christian in the first century; what you believed was not set in stone and varied between and within churches; what you practiced depended on your ethnic background (Jewish or Gentile) or which apostle you followed (Paul or the Twelve).

There are a few other things of note within early, 1st century proto-Christianity. Women appear to have had a much more prominent role than in later centuries; in chapter 16 of his letter to the Romans, Paul speaks highly of a woman named Phoebe, referring to her as a "deacon," and to another woman named Junias and her husband as "prominent among the apostles." While these titles then did not have the sort of fixed hierarchical or institutional meaning we now associate them with, that Paul used such language to describe women in his church speaks to a much more gender equal (though probably not fully egalitarian) community than it would later become. Indeed, this fact seems to have so discomforted later Christians that later impersonators of the apostle Paul, writing the pseudepigraphic pastoral epistles of Timothy and Titus, explicitly forbid women from speaking in church of holding authority over men; a line was even interpolated into Corinthians reflecting this same opinion. Nonetheless, scholars are fairly certain that women had a much more elevated role in the early Jesus movement than they would from the second century onward.

Lastly, most members of the Jesus movement in the 1st century would not probably not have had gospels to rely upon for guidance, at least not in the way later Christians would. The canonical gospels of Mark, Matthew, Luke and John would only come to exist in the latter half of the first century, in about 70, 85, 85, and 90 C.E. respectively, and how widely circulated they were by the year 100 is uncertain. Proto-Christians may have had some earlier documents like the passion narrative in Mark, or possibly a collection of quotes of Jesus as are found in the Q source or a hypothetical earlier form of Thomas. They may also have had some of Paul's letters; this is actually quite likely given a line in 2 Peter 3:16, where the pseudepigraphic author refers to Paul's letters as "scripture." This suggests that the author, writing in the first half of the 2nd century, had access to Paul's letters and expected the Christians he is writing to to be familiar with them as well, suggesting a fairly widespread circulation and acceptance as Scripture by this time. Even if 1st century Christians did have a gospel or some form of a gospel, they probably would not have had all four canonical gospels or all of Paul's letters; later writers such as Serapion and Irenaeus often speak of communities using a specific gospel rather than gospels, such as the Syrians using the Gospel of Peter and the Jewish Christians using the Gospel of the Hebrews. So what texts you would have had as a 1st century varied widely, and you would certainly not have had the full New Testament as it is today, especially since several of its books had not yet been written.