So I've already started playing through the game and beat Aqua's segment, but I was curious about this: Are there any advantages (story-wise, overall experience with the game, etc.) to playing Birth By Sleep in a particular order?
As I'm aware there is no advantage to completing one before the other (no special moves, areas, items unlocked). It's simply a matter of how you would like to experience the story.
If you want a fairly straightforward approach/unfolding of the story, I recommend playing all the way through as Terra, then Ventus, then Aqua. I also recommend you play on Proud Mode if you want to unlock the extra content as it's the easiest difficulty that doesn't require lots of grinding on minigames. This method also ensures you will simply learn each character as you go.
You could also try to have a separate file for each character and continually jump from character to character as you complete each world. The reasoning for this would be if you wanted to see the entire story in chronological order though it may simply be more confusing, story-wise and especially gameplay-wise. I have yet to try it myself but it would require a timeline to execute correctly.