The great thing about Los Angeles is the open space and hiking. There's lots that's close, and quite a bit that's kid-friendly.
If you do any hiking in southern California, it behooves you to learn about poison oak. It's basically the bully of the weed world, and there is no arguing with it. Starboy kicked some along the path, but I stripped him down before we got back into the car. If you touch any of it—leaves, stems, sticks, or even dead sticks—you have only 10 minutes to try to get the oil washed off before entertaining a nasty rash. Even on a short trail, that's not really enough time. Once the oil is on your clothes (or, as a friend found out a year later: inside your sleeping bag), you can get re-exposed repeatedly. Better to avoid it—and the dogs that obliviously play in it—in the first place.