Which person of the Holy Trinity is responsible for the continuation of the church after Jesus resurrection?

Answer 1

Gross over-simplification, but:

The 'church' (it wasn't called that, then) split into factions; Paul mentions his concern that people were following their favourite human leader, one saying, 'I am of Paul' and another, 'I am of Apollos', but he stressed their unity as followers of Christ. However, a more serious split occurred between what might be called the 'Gentile-inclusive' church, under Paul and Peter, and the Jerusalem group, strongly Judaistic, under Jesus' brother, James. The latter group seem to have faded out, or were reabsorbed into mainstream Judaism.

Later, a collegiate form grew around the Mediterranean with each Bishop preaching their own variant, some Gnostic in nature. They were eventually united in the time of Constantine. It would seem he saw great political advantage in a unified religion for the Empire; the old roman religion was a bit tired - people were joining the mystery cults, but they were too small and anarchic for political purposes; Mithraism, a strikingly similar faith to Christianity, was widespread and popular with the army - but he may have rejected it for that very reason; the army were enough of a danger to the emperors without handing them the priesthood as well. Better to knock Christanity into the desired shape. He assembled the Bishops and they voted for the doctrines they wanted; thus the Roman church developed.

There seems to be an element of truth in the idea that Thomas went to India; when Christianity was 'introduced' there by missionaries, they discovered an existing Christian church. Unfortunately, their beliefs and practices were viewed as heretical, and the adherents were more-or-less forcibly integrated into what is now (since the reformation) called Roman Catholicism. I have heard that there is a remnant of the original still existing; but haven't good info on that.

Roman Catholic AnswerJesus established His Church on the rock, Peter (Peter in Greek, or Kephas in Hebrew, means Rock - it had never been used as a name before then) as his Vicar or Prime Minister (Matthew 16:18). The apostles, in union with him, were the beginning of the Church and Jesus continued it through them promising to be with them and it until the end of time (until the consummation of the world) (Matthew 28:20).

Answer 2

All three, esentially. But more specifically the Holy Spirit, sent by Christ to as the Guide and Advocate. (See the Gospels)