I would have to concur with Imperial Angel. The Tsar’s children appear to have realized at a young age what and who they were. The Grand Duchesses seem to have been very unaffected by their position. They were given a comparatively spartan upbringing. Hard beds, simple furniture, plain food. Tatiana alone was referred to, by some observers, as being rather haughty. Given that she was very much like her mother, however, it may well be that she was simply uncomfortable with strangers, and her behavior, like her mother's, was misinterpreted.
They were born into a royal family, the life of a Grand Duchess was all they had ever known, and I'm sure it must have seemed every bit as normal to them, as our way of living is to us. It was just a part of who they were. Alexei was a bit different. Where the Grand Duchesses were never allowed to treat people in a condescending manner, the Tsarevich, because of his illness, was allowed to get away with far more. He was usually kind, but had been known to misbehave outrageously. He told several people that he could do, more or less, whatever he wished, because he was the Tsarevich. The Tsar was embarrassed when he did, but never really scolded him.