I think watching the play made Arya realize that she couldn't become no one in order to be one of the Faceless Men. That her love for her family and her desire to avenge them are things she simply can't let go of. Telling Lady Crane she should write for the play was a way of saying "be true to yourself, be who you are". Ned Stark didn't raise his children to kill those who did not deserve death simply because someone was willing to pay a price for them to do so.
To a degree. I do honestly think that watching the play, being put on the spot by the actress, then actually understanding the motivations behind the real Cersei was sort of a lightswitch moment. I doubt she forgives Cersei and wants to be BFFs, but I think she had a moment of introspection and realized that people do things for a reason. Kind of like the actors playing the role/part, people will act in certain ways and it is hard to flip the script. Kind of like the Hound. She viewed his actions and wanted him dead. Then she understood he basically had his role and had to fulfill that role and she ultimately didn't hate him in the end.
I do wonder where Arya's story will bring her now, though. There are only a few possible options currently:
1) She actually dies and the Waif takes her place (unlikely, but you never know!)
2) She beats the Waif somehow (Arnold vs. Predator style?)
3) The Waif ends up defeating herself. Arya isn't no one yet and still a trainee, but the Waif is supposed to be no one. If the Waif displays traits that show she is not no one (ie, desires), then the Waif is actually committing a much more grave offense than Arya. I actually think this option is the most likely. Jaqueen will basically take out the Waif, thank Arya for helping weed out the unworthy, but then tell her she needs to go because she clearly will not become a true faceless.
As for what happens post Bravos, my guess is she either meets up with Sam n' Crew (Old Town is near Bravos, isn't it?) or she decides to just go back to Westeros and look for her family on her own.