Where would you have taken this series after this movie?

I would have added The Joker, Selina Kyle, Two-Face, Bane, Talia al Ghul and Robin.

I'd continue the story by having The Joker tearing Gotham apart through chaos and turning Harvey Dent into Two-Face. Batman would win based on a major lie, and Bane would soon rise up to lead the League of Shadows back to their former glory in the place of Ras al Ghul. I think it woulda been cool. Oh well.

I would have kept the same visual style and went for other villains that were less seen in films and television in the second film. Two-Face and Joker for the third, after establishing and developing Dent.

And I wouldn't have made the League and Ra's be the "big bads" of the whole series.

I'm fine with where TDK went but Bruce shouldn't have given up being Batman for 8 years. From TDK onward, in my version of events, he would go on being Batman but would have to contend with the police being after him at every turn while he fights crime, and then eventually the truth about the murders would come out.

Make Selina Kyle the female lead in all three films. Rachel Dawes was probably a two scene character that was forced to become the female lead for two films.

The trilogy was like shorten to two films but became three films again, Batman retired, came back and retired again.

Begins would've included Dent. And there you would see a beginning of Bruce/Harvey friendship. The sequel would of course include The Joker and Harvey Dent, but Rachel wouldn't be romantically involved with him, cause she would understand that Dent in reality is an unreliable, unpredictable, impulsive attorney obsessed with fighting criminals (especially the crime boss Salvatore Maroni, secretely he'd like to eliminate him personally) through quasi-legal means. The Joker of this movie would be a completely different Joker, who wouldn't be a political activist trying to make a point, he would just want to have fun by killing people, trying to outsmart Batman (And Maroni would be grateful for that, Joker's activities would be a great distraction turning Batman/Gordon's attention away from the mob's businesses.) But Dent wouldn't care about the Joker, he would be obsessed by the mob, not by some clown. And this is the main difference between this proposition and what ended up in the final movie: movie's two major components would be: the first - Batman/Gordon trying to catch the Joker, and the second - Maroni/Dent conflict. In the end, the Joker would abduct Maroni, threatening to kill him... Batman would try to stop him, but Dent wouldn't. Ultimately Batman would stop the Joker, who would later escape, cause Harvey would enter the scene to end Maroni's life...

It's only a sketch, but it shows changes I'd make to improve the story of TDK, which I don't really like.

The Dark Knight would have taken place midway in Batman's career.
Not 1 year after Begins.
There would have been a lot of hints and occasional discussion of what had come before.
Dick Grayson would have already come and gone.
I would have had cameos from other villains similarly to Scarecrow.
Harvey Dent would have already known Bruce and Gordon.
His relationship with Gordon would have been more cordial.
Rachel would have been dead from the start. There was too much focus on this character.
I would have liked to see the movie overall more like The Long Halloween.

Dark Knight Rises is near perfect.
Again there would have been more of a passage after TDK.
I would have made Gordon older. He seemed too young still.
I would have changed John Blake's name to Tim Drake.

I loved the trilogy but there are a few things I would've done differently. I would change the batsuit in TDKR. It's been eight years, Bruce should've upgraded it to be. Just for superficial reasons because I wanted the more bulky panther look from BB to come back. They could've done something like that but make it more flexible. The last two movies felt less like Batman. In Batman Begins, Gotham city was perfectly depicted and Batman was always in the shadows and acted more intimidating. He was like that as well in TDK but not that much and not at all in TDKR. I would also add more villains that can be Nolanized like Penguin and Riddler.

EDIT: After Arrow, BvS and the Telltale Batman game, I really think a voice modulator would've suited this version of Batman. We wouldn't get people saying he had throat cancer as well. This guy on youtube made a video where he changed Bale's voice to make it sound more like Affleck's and it sounds awesome.

https://youtu.be/xFhgHaCaaGg