Women can wear just about anything. A cocktail dress in a comfortable fabric is ideal because it connotes respect for the performance yet allows the user to move freely.
Good clothing choices for the ballet are items that are comfortable yet look good such as pair of khakis and a button down shirt.
Attending the ballet for the first time can be very exciting for a little girl or boy. By all means emphasize the hard work of the dancers by dressing a small child in their finest clothing to attend a dance performance.
Matinees are performances given in the afternoon. A matinee tends to be a less formal occasion than a later performance. A short cocktail dress or a casual pair of slacks is appropriate for this occasion.
Evening performances tend to be more formal occasions so you may want to dress up in formal clothing such as an evening gown or tie and tails. A shorter cocktail dress or neatly pressed pair of pants is also still appropriate.