I know the really basic history of that period, Julian the Apostate, the Ecumenical Councils, that type of stuff. But, I read one thing that interested me the other day, that the Altar of Victory, an alter to a Pagan goddess, was still present in the senate until 357 (returned by Julian and removed again 382). Of course, it would be unreasonable to expect that the moment Constantine converted to Christianity they would tear down or convert every single vestige of Paganism in the entire empire but it seemed odd that it would take them 50 years to remove such a protonate aspect of the old religious order. Is there a reason why this remained in the senate for so long? Where many (most?) of the senators still Pagan during this period? Did the Christians decide that it was now only symbolic (like say lady justice)?
There are several different events that you've kind of collapsed into one here. Between Constantine's conversion and the empire's conversion there were nearly 60 years of history.
312 BCE: The Battle of Milvian Bridge. Theoretically, Constantine became really interested in Christianity at this point after a religious experience at the Battle of Milvian Bridge, so arguably his conversion begins here. We don't hear about this until years later, though, through Eusebius which is a questionable source.
313: Edict of Milan. Licinius writes to Constantine affirming that Christianity will become legal, as in no longer persecuted (it's assumed that Constantine wrote him back and was down with that, but we don't have record of it). This is in direct response to a recent history of the Christians being persecuted under Diocletian. Constantine himself is ostensibly Christian at this point, and some argue he is studying it, though he has not officially been baptized.
325: The Council of Nicea. A unified version of Christianity is recorded and made "official" within the Catholic church. It's not the official version of the empire, though.
337: Constantine is baptized on his deathbed and dies.
So at this point Christianity was legal and Constantine was officially Christian, and his son who succeeded him was Christian as well. But it still was not problematic for citizens to not be Christian and the government was not Christian, either. From a polytheistic/pagan perspective, this was pretty normal; the Romans had assimilated many religions into their own.
380: The Roman Empire converts to Christianity. Theodosius I, who is emperor some 9-10 reigns after Constantine, makes Christianity the official religion of the empire.
Hope this helps!