Contrary to what people may tell you, the evidence for the various ancient liturgical practices in Christianity is pretty substantial. We have several witnesses to the continuity of the liturgical practice from people like St Justin Martyr (~130 AD, First Apology, Ch. 67
here) and Hippolytus of Rome (~215 AD, you can read his Apostolic Tradition
here). Among other sources, of course. When you read these two in particular, though, you'll find that not only the original form or outline of the service has remained unchanged since the 2nd century but in many cases the prayers are word for word identical to what is served today.
We have also historical evidence in the form of archaeological findings. The earliest house church we have found is in Dura-Europos and dates to around 235 AD. This church contained a baptistry and icons, and parchment fragments that match the Eucharistic prayers of the Didache (chapter 9,
here). Ancient churches like this and the one at Megiddo also had places for the celebration of the Eucharist.
The very most basic history is that the earliest Christians were Jews and gentile God-fearers. As such they went to Synagogue on the Sabbath day, and this is where St Paul was accustomed to going first in each city. The followers of Christ who would become Christians would also meet on the Lord's Day (Sunday) for the love-feast, the Eucharist, Communion. The earliest description of the Church in Acts 2:42 says that those who believed devoted themselves to three things with definite articles:
the teaching (didache) of the apostles, and in fellowship (koinonia, communion)
the breaking of the bread, and
the prayers. When the Christians were put out of the Synagogues this worship practice continued, and they more or less smooshed the two together: the liturgy of the word, and the liturgy of the faithful.
The former is incredibly similar to synagogue services, which should be no surprise as that is where it is derived from. The latter is uniquely Christian, and culminates in the celebration of the Eucharist. Incredibly ancient practices such as the chanting of antiophonal psalms and hymns and the words themselves are no different than they were practiced by Jews and Christians in the first and second century. The entrance processions in the Divine Liturgy correspond to synagogue practices which are maintained today in Orthodox Jewish synagogues. The agape meal itself is ingrained in the Orthodox Church practice of the after-service meal - affectionally known as "Coffee Hour."
There are several very ancient liturgies - the Liturgy of St James, and the the Liturgies of St John Chrysostom and St Basil the Great. They all pull from extant practices. It seems likely that the form of the liturgy we celebrate today (St John Chrysostom most often, but also St Basil ~10 times a year) was a combination of a more monastic tradition with the cathedral tradition in the cities. This is things like which prayers are said when, whether some parts are done in silence vs with prayers or chanting, and so on. You can read more
here.
The Latin rite is a translation and has its own history. There are also other liturgical practices such as vespers (the oldest hymn we know of is
Phos Hilarion, O Gladsome Light, which is sung at Vespers and was referred to as an ancient in the 4th century) and the Presanctified Liturgy of St Gregory the Dialogist. All of these have ancient roots, and a pedigree that is a matter of historical fact.
Fun note. The Latin rite is named from the word "missa" in the dismissal, "Ite, missa est." Which means "go, it is finished" or "go, it is sent" or "go, it is the dismissal."
