What is the correct sequence of evolution of the brain?

Answer 1

First, there is no "the correct sequence". There is the most likely sequence of key historical developments as best as can be established by science.

Second, there is no "the brain". There is a wide variety of more or less complex central nervous systems, most of which have a focal node of some sort that can be called 'brain'.

The simplest of such central nervous system, and most likely the first type to emerge, is no more than a chord of nerves along the length of the organism, with or without ventral extensions, terminated by a 'knot' of nerves forming the 'brain'. Such simple nervous systems can still be found in flatworms.

Arguably the most complex brain known to us is the human brain. In the structure of this brain we can see something of the evolutionary development of brains in our lineage: layered from the basic 'reptilian brain' to the most recent development, the neocortex. (Note that referring to these layers as the "reptilian", "paleomammalian", or "neomammalian complex" can no longer be considered scientifically accurate, given that recent findings have pushed back the development of complex behaviour and associated neural structures much further back in time.)