You can convert in the faction screen, under the summary tab. To convert to a certain religion, at least 35% of the population has to follow that religion. In addition, you get a +10 attitude with every faction of the same religion and a -10 attitude with every faction of a different religion For the details, see the individual religion pages
The state religion gives a unique factionwide bonus, and gives access to a religious building chain and religious edicts.
