This is a really cool question right here. This particular subject, the progressive conversion of the Roman Empire to Christianity, has been widely written about by historians from various countries. Constantine himself did show interest for monotheistic ideas even before the date of 312, I'll not go into details here as it is not the suject of this thread but the Panegyrics of 307 and VII 310 are really interessting to show his early attraction to specific deities, Apollo/Sol Invictus in this case. Starting from here you can understand that Constantine, before becoming a christian, was himself very aware of the importance of pagan traditions to assert his power.

Constantine was a man of compromise, he did not use threat of violence to impose Christianity in the Empire. I'll tried to gather the information I gathered a few years back about the coinage during his reign (which perfectly show this compromise) but I also like the idea of T. Bauzou I read a few years back in his book De La Cité À L'empire : Histoire De Rome about the very symbolic signification of the Chi Rho symbol used during the battle of the Milvian Bridge. According to him the X crossed with a P is a well thought compromise to satisfy both eastern/greek-speaking/christians that could see in these two letters (XP) the word « Christos » (Greek for God : XPICTOC). On the contrary, western/latin-speaking/pagans would most likely read the letters 'inversed' (PX) as the word PAX (peace in latin). I have not seen this idea being talked about a lot though. I would most likely take some flak from my fellow historian collegues here, but eh, I like the idea as it shows the use a pagan tradition adapted to Christians.

As for the coinage, which always was the main vector of propaganda during the Roman Empire, we can notice different things. During the period of 312-324, mints remain ambiguous on the religious level. We can see that most of the coins until 320-322 - 324/5 are consacred to Sol Invictus, being the invincible companion of the Emperor. This coin (313/317) is a typical example of Constantine pagan coinage. The coin shows Sol Invictus half naked, wearing the solar corona (solar emblem par excellence, symbol of the universal and eternal sovereignty.) The globe in his left hand (military symbol of universal domination) and right hand raised (palm open in a sign of greeting and blessing). The SOLI mention INVICTO COMITI 'the undefeated Sun, my companion' is part of the pagan tradition that the Sun is to the Emperor a protector companion. The invincible God, supreme and eternal, which was established under Aurelian in 274, and that supersedes the pagan pantheon guarantees the political legitimacy of anyone posing as Augustus at the time.

This coin now (Solidus Ticinum 320), shows the Sun God, with all its attributes, given a Nicephore globe (globe of Victory). We see spring of this globe a character that will crown a laurel wreath Constantine standing right. This part also includes the words SOLI COMITI AUG (USTI) N (Ostri) "Sun companion of our Augustus" is a new solar formula. Some historians, including A. Alfoldi, place this as one of the last solar reference under Constantine (324/5/6). As for the meaning of the new formula, Kurt Aland, a german theologian, wrote that this was a way for Constantine to reassure the pagan upper class of the East at that time. After 317 currency issued with SOL are very rare until finally disappearing with the victory of Constantine over Licinius in 324.

This other coin (Solidus Ticinum 313) made after Constantine's victory over Maxentius is very interessting I think. It shows the intimate relationship the winner with Sol Invictus. Sol Invictus, in the background with a solar corona, seems to have with him a spear or a sword while Constantine is labeled with a shield on which we see the solar chariot (the sun chariot, pagan figure, had four called horses Pyroïs, Eoos, Aethon and Phlegon). There is a certain complementarity that is put forward by the imperial propaganda. We know that the day after his victory, 29 October 312, Constantine entered Rome as a winner but we still don't know (as far as my knowledge goes) whether he has made the pagan rite reserved for the winners entering Rome.

The pagan Constantine coinage is interresting as you can see, it shows this compromise between pagan tradition/Christianity. Last coin that I will link here (I promise) (Ticinum 315) : We can see the Chi Rho symbol on his helmet + one possible cross on his shoulder (?) (would need approval on this one). Nevertheless this faith is very personal because it is he who carries the symbols, not the coin itself. Showing perfectly that the Emperor is a Christian in a pagan empire. Don't forget that coinage is a public institution of the Empire and thus must retain pagan appearances. As Paul Veyne wrote perfectly on this subject regarding Constantine 'The coinage remains so reserved in religious matters that it proves the absence of religious propaganda of the political powers.'

During this part of the reign, Constantine will take a serie of decisions that will place him in the pagan tradition of the head of state while promoting and protecting Christianity. As can be expected, after the victory of 312 the Empire still remains pagan, however Constantine will conduct a genuine pro-Christian policy. Public pagan worship did continue under Constantine. He was the great pontiff of Rome for instance. He also placed his Dies imperii (anniversary celebrating his rise to power) under the patronage of Sol Invictus placing this festival on December the 25th (winter solstice). Constantine also did declare Sunday "Day of the Sun" on March 7 321/325. A day that will become a holiday, which is why even today some cultures have kept the Day of the Sun for Sunday: Sunday (in English) Sonntag (in German). He also gave back to Christians their confiscated assets during the persecutions. He built in 320 a church for the bishop of Rome, the current St. John Lateran. He also gave christian churches the right to receive 'inheritances' same as the pagan great temples. The money allocated to the Church was an intent to put the latter only on an equal footing with the pagan temples. It is important to note that Constantine simply gives Christianity the same privileges paganism already had in his time.

At last, the Edict of Milan in 313 is the perfect example of this equality sought between the two religions. The text is only a renewal of the Edict of 311 (edict of Sardis) that allowed Christians to assemble and to worship. The new text now contains the return of books and buildings. It is published in Nicomedia in 313 via circular letter by Licinius and promotes universal tolerance rather. The text called Edict of Milan, allows everyone 'to worship in their own way the divinity that is in heaven.' It gives Christians freedom of worship, equal rights and the return of all confiscated properties.

 

EDIT: Most certainly bad english/typo.

 

EDIT2: I will not go further into depth as this post is already quite long, but I'm sure other people will explain to you other aspects of Constantine reign which are also very interessting.