The Blue Exorcist anime follows the manga to a considerable extent. Some minor details might be different for better animation flow, but the core story remains intact.
The Blue Exorcist anime is a decent adaptation of the manga. It keeps the core elements intact, but there are added scenes and tweaks here and there to make it more visually appealing and engaging for the viewers.
Yes, the Blue Exorcist manga often progresses beyond what's shown in the anime. The manga has more detailed storylines and character developments that haven't been covered in the anime adaptation.
You can start from the chapter that follows the events of the anime's conclusion. Usually, it's a good idea to look for recommendations from fans or online manga forums for the exact starting point.