I stand to be corrected on this, but I don't give too much credit to the idea that culture, gender or even age (over the age of 13, at least) play a greater role in a character's outlook than their own inherent character traits. The quiet, subdued, but kind parish pastor is probably far more different from his raucous, impulsive and violent parishioner than from the equally quiet and kind keeper of the new Hindu temple across the street, or from the equally subdued leader of the women's Bible study group at the church. In fact, the dynamic between the pastor and the parishioner may well be reflected in both of the other contexts.
In other words, the difference between people of different ages, genders and often cultures is not one of inherent character, but one of situation. Of course the Saudi Arabian woman's enforced seclusion by virtue of the culture she grew up in has affected her character, but it is the situation, not the culture that created it, that has affected her; an America girl would experience the same effects were she to be put in that situation. Likewise, growing up during the Cold War would affect an older character, but only in the same way it would affect me if it has continued into the 1990s and 2000s. All you need to do is understand the situations, challenges and encounters the character's context and attributes produce, and you can work out the rest in the same way you would work out any other character.
The one major exception to this is culture; different cultures often have different outlooks on many things. That said, these differences are often exaggerated; the only ones likely to come up in a story are social and relational differences, particularly how people interact with their elders, their father and their mother. Research is your only resort here, I'm afraid.
In terms of establishing character, vocabulary is essential. It will always come up in dialogue; if you're using a first-person narrator, it will permeate the narrative as well. You can even use it subtly with "tight" third-person narrative (I think the technical term is "limited third-person narrative").
How to write for different characters is about twenty different questions, depending on what sort of character you want to write. A few have been asked here before, and once you have an idea for a specific character, you can always ask a question about that character specifically.
Writing a "child" character can be split into two categories: Over-thirteen and under-thirteen characters.
"Children" can be surprisingly adult over or even at the age of 13. The main difference between teenagers and adults is their social context, how they interact with each other and with people of different age groups. Again, a little research, even if it's just talking with a group of teenagers, will go a long way in helping you to characterise them.
Teenagers are also, of course, known for rebellion against authority, and young characters are often naive, but there's no real difference between these traits and the rebelliousness and naivete that some adult characters might possess. And, of course, their vocabulary and use of language is likely to be different to any other social group, but again, it's a matter of research.
I'm afraid I can't in good faith tell you anything about writing characters under the age of 13 except that I suspect it would be incredibly difficult, especially as the viewpoint character. I'd imagine it's rather a large topic, so you might want to ask it as a question on its own, where hopefully someone can give a better answer than I.