I don't really feel like writing a new answer, so here is an old one of mine that covers much of this terrain. Happy to answer follow up questions here though.

I think the first thing you have to consider is that it wasn't just binary - first Christians were hated and then they were loved. The Roman Empire at that time was huge and widespread and there were all kinds of cults, minor religions, and varied belief systems throughout the Empire. Also Christianity at that time was a amorphous thing - it wasn't like Christianity today with the various denominations all of which have defined belief systems and statements of faith. A follower of Jesus Christ as an Early Christian could be practicing vastly differently from someone who lived 100 miles down the road. Prior to the Council of Nicaea, there really wasn't a "standard" form of Christianity.

So within that framework, the watershed event for the Roman Empire was when Emperor Constantine converted to Christianity and declared himself "Emporer of the Christians". There's a lot of question about why/how he converted. He was likely influenced by his mother and there are sources that will talk about him having a dream or a vision (or both) that told him he would be victorious in battle if he followed the Christian God. He did win several key battles while fighting under a Christian flag, and that kind of cemented things under his rule. After that he became a patron of sorts - building basilicas, promoting the study of the faith, returning confiscated property, etc.

(A lot of Christians consider the conversion of Constantine a miracle, because it was THE conversion that led to Christianity becoming a major religion.)

Interestingly, Constantine didn't force his people to convert and in fact under him the Roman Empire issued edicts that provided for the rights of the people to practice any form of worship they wanted to and stated that the government could not force a religion on anyone.

Later on things changed and Christianity became the official religion of the Roman Empire. It actually did happen fairly suddenly, though. Once Constantine converted, it was only a matter of maybe a decade or two before Christianity became the primary religion of the Empire.

Eusebius' The History of the Church from Christ to Constantine is sort of the definitive text on the subject, written by a contemporary of Constantine's.