The Christian mass developed out of the Love Feasts mentioned in the New Testament in verses such as 1 Corinthians 11:20–11:34 and 2 Peter 2:13–2:13.
We're lucky because these love feasts show up on a few early Christian frescos so we actually know what they looked like and how they were ordered. Here's a good index of those feasts.
These feasts, as best as we can tell, appear to be a reenactment of sorts of the Last Supper. They started with the breaking of bread, and concluded with the sharing of wine. It was all rather informal.
It must be emphasized that at this early phase of Christianity, The Love Feast was not a form of worship necessarily. These people were still Jews. They still gathered on Saturday to worship. On Sunday they would then gather to ponder Christ together. This difference is actually also offhandedly mentioned in the scriptures as well, in Acts 20:6-7:
And after the Feast of Unleavened Bread, we sailed from Philippi, and five days later we rejoined them in Troas, where we stayed seven days. On the first day of the week we came together to break bread. Since Paul was ready to leave the next day, he talked to them and kept on speaking until midnight.
You see here that the Early Christians were not divorced from the regular Jewish worship schedule. Sundays was not a form of worship. It was a gathering to discuss and debate about Jesus' teachings around a meal, the day after worship. This might be because they viewed this as a work and so didn't want to do it on a Sabbath.
You once again see a prototype of the Christian Mass in 1 Corinthians 16:1-2, when along with the above, donations start to be given to the church on the first day of the week:
Now about the collection for the Lord’s people: Do what I told the Galatian churches to do. On the first day of every week, each one of you should set aside a sum of money in keeping with your income, saving it up, so that when I come no collections will have to be made.
From this we can get a sense that Sunday was increasingly the day Christians would gather around a dinner meal to discuss Jesus and manage their communal finances, and Saturday was the day they rested and went to temple to worship.
What happens to evolve these elements into a unified service is a bit unknown, but by the time of Origen in the early 3rd century, this informal gathering of food and debate and financials was formalized into several liturgies, and functioned as a kind of day-long worship service on Sunday. Origen makes mention of a few of these formalized practices but it's also clear they have remained mostly disorganized and are merely individual practices that occur throughout the day rather than at a single hour-long church service. I'll post a few passages of his liturgical musings that show some resemblance to what Christians do today but at this time were more informal.
There are things among the Church's observances, which everyone is obliged to do, and yet not everyone understands the reason for them; e.g.,... the fact that we kneel to pray, and that of all the quarters of the heavens, the east is the one we turn to when we pray... And can you readily explain the reason for the way we receive the Eucharist, for the rites it is celebrated with or for the words, gestures, commands, questions and answers made in Baptism?
...
There is no limit to the number of postures the body can take up, but the position to be preferred is unquestionably the one we adopt when we stretch out our hands and lift up our eyes, as it is the best bodily expression of the soul's attitude in prayer.
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I say that this should be observed when there are no obstacles. But circumstances may sometimes lead you to pray sitting down, e.g., if you have... bad feet; and if you have a temperature, you may even have to lie down... for if your business makes it impossible for you to go to some quiet place to discharge your debt of prayer, you will not be able to insist on standing when you pray.
...
And here, we see the first inclanations of the idea of a church building. Origen is laying the foundations of what will eventually be the argument for gathering in a specialized space dedicated to prayer just like the temple of the Jews:
As far as place is concerned... any place will become suitable for prayer if you pray well in it... How-ever, if you want to say your prayers in greater quiet and without so much distraction, you may choose a special place in your own house, if you can have a consecrated place, so to speak, and pray there.... Special grace and benefit are to be had from the place of prayer, the place, I mean, where the faithful assemble; for it is reasonable to suppose that angelic powers are present when the faithful meet together; the influence of our Lord and Savior must be there too and so must the spirits of the saints - the spirits, to my way of thinking, of the dead who have gone before us and obviously, too, the spirits of those saints who are still alive, though how, it is difficult to say.
Fast forward a mere century or so to St Ambrosia, and we see these articles of Origen have formulated into regional formal liturgies. We know there were regional variations because in writing to St Augustine, St Ambrosia says:
When I visit Rome, I fast on Saturday; when I am here, I do not fast. On the same principle, do you observe the custom prevailing in whatever Church you come to, if you desire neither to give offense by your conduct, nor to find cause of offense in another's.
Ambrosia himself has the formal Catholic Liturgy attached to his name, through we cannot say for certain if he invented some kind of official liturgical worship we today would call the Mass. But we do have stories of what sounds like him inventing the model of the mass we use today to keep refugees focused on pray in times of fear and worry. Henry Jenner writes in the 1913 Catholic Encyclopedia:
We know from St. Augustine (Confess., IX, vii) and Paulinus the Deacon (Vita S. Ambros., § 13) that St. Ambrose introduced innovations, not indeed into the Mass, but into what would seem to be the Divine Office, at the time of his contest with the Empress Justina for the Portian Basilica (on the site of San Vittore al Corpo), which she claimed for the Arians. St. Ambrose filled the church with Catholics and kept them there night and day until the peril was past. And he arranged Psalms and hymns for them to sing, as St. Augustine says, "secundum morem orientalium partium ne populus mæroris tædio contabesceret" (after the manner of the Orientals, lest the people should languish in cheerless monotony); and of this Paulinus the Deacon says: "Hoc in tempore primum antiphonæ, hymni. et vigiliæ in ecclesiâ Mediolanensi celebrari cœperunt, Cujus celebritatis devotio usque in hodiernum diem non solum in eadem ecclesia verum per omnes pæne Occidentis provincias manet" (Now for the first time antiphons, hymns, and vigils began to be part of the observance of the Church in Milan, which devout observance lasts to our day not only in that church but in nearly every province of the West). From the time of St. Ambrose, whose hymns are well-known and whose liturgical allusions may certainly be explained as referring to a rite which possessed the characteristics of that which is called by his name, until the period of Charlemagne, there is something of a gap in the history of the Milanese Rite, though it is said (Cantù, Milano e il suo territorio, I, 116) that St. Simplician, the successor of St. Ambrose, added much to the Rite and that St. Lazarus (438-451) introduced the three days of the Litanies.
So we can definitvly put the start of the formal mass as we know it today to around the year 400AD.
It's interesting to note that by and large most of what we would call formal Christian worship can be dated to a few decades prior to the year 400AD. It seems in these decades, the Christians were heavily contesting with the Arians. Ambrosia himself inherited an Arian bishopric. And this is just speculation, but it seems unlikely to me to be mere coincidence that the first time Christmas is officially celebrated on the 25th of December correlates to Arius' death. It appears to me, that the Arians are really what forced Christians to formalize a unified liturgical mass. Why that is I can only speculate. Perhaps the Arians had a more unified Liturgy so the Catholics wanted to mimic that. Or perhaps the liturgy was a means by which to emphasize Catholic Christology and route out Arianism in the worship service.
In any case, bu the year 400 the mass is more or less in a typical western form that can be called a "western rite". I chose to focus on this rite as it's what I'm more familiar with. It's worth noting this didn't get adopted by everyone, and the East had its own distinct litrugy. India, too, developed its own St Thomas liturgy which unfortunately I do not know much about other than it exists. Also, up in Britain a distinct liturgy developed as well and would continue to exist until the Norman conquests. There is an amusing letter by a Bishop who was getting frustrated by this local liturgy which was very different from Rome's. He writes:
Britones toti mundo contrarii, moribus Romanis inimici, non solum in missa sed in tonsura etiam (Britons are contrary to the whole world, enemies of Roman customs, not only in the Mass but also in regard to the tonsure)
The author of this was either Saint Aldhelm or Gildas around the 7th century. So we can say for sure that regional variations continued for centuries after the Arian controversy.