I am trying to learn when did things start to go wrong in Christianity and when did it start to change from a peaceful, loving religious group into a movement that engaged in war and physical repression, including executions, in the Middle Ages. Therefore I was wondering who was the first person killed by a Christian, two Christians or a group of Christians due to religious disagreements.
Your question starts with a couple of assumptions that don't hold up. First, the earliest Christians had plenty of heated disagreements, starting with doctrinal conflicts over the extent to which the Law of Moses was binding (or not) on Gentile believers. Chapter 15 of Acts, and the whole of Paul's letter to the church at Galatia, record those issues. Second, even at the peak of the Inquisitions, only a very small percentage of accused heretics were sentenced to death.
(Please note, I bring these up not to justify anything -- even one killing by the church for heresy was one too many. I just want to correct some common misconceptions, ones that I used to share until I dug more deeply into the facts.)
That said, the specific you're seeking, THE first person to die for heresy, may be a detail lost to history. Based on my own very limited research, the earliest major, fatal action against a heretical movement was the persecution of the Donatists during the time of Augustine:
It seems there were strong political factors involved, not merely the suppression of a heresy. And according to the article, even though Augustine argued that the use of force was justified, he later protested the extremes that the Roman authorities were going to.
So to sum it up, TL;DR style: the use of violence against heresy seems to have come about in conjunction with the church becoming a political force rather than a strictly spiritual organization.
Religions are composed of people, and people are sinful and foolish. You'll find all kinds who identify as Christian or otherwise, good and evil. Your query is, thus, fundamentally flawed. There was never a transition point where Christianity became a "full fledged killing machine." Christianity, along with pretty much any other ideology, has at times been used to justify horrific acts, and at other times has been used to motivate man to be the best he can be. Just look at the late Reverend Martin Luther King Jr.
But if you are interested in the first recorded deaths that occurred in relation to Christianity, you'll find those in scripture:
Acts 5:3-5 Then Peter said, “Ananias, how is it that Satan has so filled your heart that you have lied to the Holy Spirit and have kept for yourself some of the money you received for the land? 4 Didn’t it belong to you before it was sold? And after it was sold, wasn’t the money at your disposal? What made you think of doing such a thing? You have not lied just to human beings but to God.” 5 When Ananias heard this, he fell down and died. And great fear seized all who heard what had happened.