I think that's why they had Zuko trekking through the Earth Kingdom in season 2.
Azula was always a bit sadistic for a child, but you were right about how they were brought up to believe that the Fire Nation was superior.
But its interesting to note that when Zuko was a fugitive during that season along with his uncle, he ended up growing out of that mentality. He saw the effects of the war first hand, traveling on the run and spending time with that family whose son had gone to fight in the war, and how angry they became when they found out he was Fire Nation- even the kid he befriended who saw him like a brother ended up hating him.
What we are taught to believe could be undone when we learn to see first hand what the right thing really is. I think that's what made his transformation so different. Azula never escaped that mentality of trying to be seen as the dominant offspring in her fathers eyes and she would never stop clawing until she reached it.
But you were right, the episode "The Beach" really did paint an oddly human picture of her. She liked boys, she was awkward around them, they was sorta shy and found it difficult to transition from fierce Azula to normal Azula. That whole atmosphere was the entire essence of her being. She was arguably one of the best TV villains in a long time and seeing her pull strings and make things work was impressive. Additionally, she conquered Ba Sing Se in a matter of 3 days with only her two friends. Her uncle attacked the outer wall for over 600 days with an entire army at his back and he failed to claim it.
She was a masterpiece in writing and voice talents and she'll never be forgotten for her iconic role in the show.
Also, her speech to Long Feng was BEYOND phenomenal. Azula: "I can see your whole history in your eyes. You were born with nothing. So you've had to struggle and connive and claw your way to power. But true power, the divine right to rule, is something you're born with. The truth is: they don't know which one of us is going to be sitting down on that throne, and which of us is going to be bowing down. But I know, and you know." Long Feng bows down: "You've beaten me at my own game" Azula: "Don't flatter yourself, you were never even a player."