Scout does not accept the world as she sees it. For example, she wonders about Boo Radley. Boo signfies the potential of people to be good or cruel--the potential of childhood. Scout also does not accept Walter Cunningham's situation as a given. She asks her father about poverty, trying to comprehend the different social classes represented. Finally, Scout is curious about the trial and the town's treatment of Tom Robinson, though she does not understand it the way Jem does. Jem does not belive that the jury will convict Tom, but Scout just isn't sure. Not being quite old enough to grasp the nuances of what is happening, she locks on to the foundational points.
Yes, certainly the fact that she is constantly getting into trouble indicates a high level of curiosity. Likewise, as is shown by her desire to find things out, such as the story surrounding Boo Radley, her character and her actions all point toward a child who is desperate to find answers to the questions she has.
It seems like Scout is always in trouble for something, and it's certainly due in part to her avid curiosity. As a little girl, before we meet her, Scout sat on her father's lap and learned to read by watching and listening to him. When we meet her, she has a nearly insatiable curiosity regarding Bood Radley. Jem does, too, of course, but he grows out of it; Scout does not. At least not until she finally meets him.
As a precocious child, Scout is naturally curious. Having a father who has a very open relationship with her also contributes to her inquisitiveness. For instance, Scout does not fear asking him what rape is and repeating what others have said about Atticus "defending n----s."
Additionally, Scout displays curiosity in Chapter 12 in which Calpurnia takes the children to church:
"Cal," I whispered, "where are the hymn-books?"
"We don't have any," she said.
Later, Scout herself narrates,
"My curiosity burst: 'Why were you all takin' up collection for Tom Robinson's wife?"
When Scout is told that Tom's wife can't go to work, Scout asks,
"Why can't she take 'em with her, Reverend?"
Reverend Sykes has to explain that no one will hire Mrs. Robinson. Later, Jem and Scout ask Calpurnia which books she used when she taught Zeebo, her eldest son, and Calpurnia answers that she used the Bible and a book granddaddy Finch gave to her.
Scout's curiosity is first shown in the shining something she sees in the knothole on the Radley's lot. When she grabs that piece of gum, sniffs it, puts it in her mouth and doesn't die, this curiosity is clear.
She shows curiosity at the jail the night she steps in the middle of a mob of men because she wanted to know what Atticus was up to.
She demonstrates curiosity when the kids watch the town go by on their way to the trial. Remember, the kids referred to the procession past their house as if it was a parade.
Finally, she demonstrates curiosity when she goes to the trial. The kids all wanted to go to see what was going on. This was more than just something to do because Atticus specifically didn't want the kids going downtown that day.