You did nothing wrong really. The point of the game is to gain power as one of the great Roman factions, but then grow too powerful for even the senate to control therefore throwing the Roman factions into civil war.

I play mainly mods, but when I played Vanilla (R:TW unmodded), I found it did take a while to start the civil war to start. I was ready, but for some reason it just wouldn't happen.

I really don't have any more advice, but welcome to the forums. Have a wonderful time!