In saying this quote, Dorothea is indicating a preference of hers in which she differs with her daughter April. April likes "old stuff," like the antiques and second-hand items she finds in the Professor's store. Dorothea has no use for these things. She likes things "new and shiny."
The quote has significance in that it points to a basic difference between April and her mother. Dorothea is consumed by glitz and glamour; she is very concerned with the appearance of things, the way they look on the outside. April at first seems to prefer the same things as Dorothea, with her fake eyelashes and attempts to appear sophisticated and worldly. There is another side to April, however, that has more depth than her mother; April only focuses on shallow values because she wants to be like her mother, and win her love. April, however, when she allows herself to be herself, is much deeper in nature than Dorothea. April has a side that can appreciate things with character and history, while Dorothea only sees what is on the surface.
The quote has another significance as well. The fact that Dorothea admits that she prefers "new and shiny" things indicates that, when things get old, they lessen in value to her. Dorothea gets tired of things very quickly, and is always in pursuit of new possessions and experiences. Sadly, one of the things she gets tired of is her daughter April, whom she leaves with April's grandmother while she looks for new stimulation in her life (Chapter 2).