So long as you don’t go off trail, shorts are fine if the weather is warm enough. You’ll want to make sure the kids use sunscreen and insect repellent on their legs, though.
Denim jeans are great during dry weather but become uncomfortable when wet. Because of this, some hikers prefer to wear nylon pants. I prefer not to take kids hiking when rain is falling, but trails in some areas of the nation – the Pacific Northwest for example – will be wet regardless if there’s a shower.
During winter, wear long johns, preferably of a moisture-wicking synthetic material. Wool pants or trousers that are water resistant would go over them.
Read more about day hiking with children in the guidebook Hikes with Tykes.