1. Teenagers like to go on dates to shopping malls and parks sounds a little more natural to me. "out on" is a little redundant, and you go on a date TO somewhere, usually.
2. Using "become" in this sentence is a little puzzling. When do they become this? On the weekends, shopping malls become the place that teenagers hang out, for example shows that it happens every weekend. Do you mean that five years ago, they went on dates somewhere else? Then it's "have become."