What were people supposed to do with an incomplete or nonexistent Bible? If god wants everybody to be Christian and the Bible is the central text of Christianity, what were people supposed to do before it was completed, much less started?

Also, is there anything on record speaking to reasons why the particular era that the Bible was written was chosen (specifically OT)? It was a gradual roll-out, so were there parts that weren't needed until they came up, or did they perhaps correlate to certain events?
The entire old testament was available and written before Jesus' time.  The new testament, as we have it is the collected writings of the apostles.  Before all the writings were complete, the apostles taught directly.  After their death, their immediate successors taught what they had learned and during that time the writings were collected, copied and distributed.

Creeds and hymns formed a big part of conveying biblical doctrine before a printed bible was widely available to all, as did the church teaching and structure.  The documents themselves were available to clergy from very early times, soon after the apostolic day.

So at first, there was the testimony of direct witnesses.  Then there was teaching from those taught by the apostles and the apostles written records.  Finally those records were compiled into the bible, and between the mid 300s and late 600s AD the canon of authoritative scripture was considered closed by most traditions, although the Catholic church did not formally declare the bible canons closed until the council of Trent in 1546 with the only disagreement with other traditions being the OT deuterocanonical books (and bear in mind, too, that the Catholic church (and perhaps others), holds to a continued revelation through the church's magisterial office and authority).