Look at some of the shade perennials that do well in your area, starting with azalias, and check to see which of your trees, if any, take to understory planting- it may be none, in which case the shrubbery will serve as part of a border between the play area and the trees. You might consider making this area of the yard into a night garden, with white flowers, silver foliage, and uplighting at the base of each tree.

It looks like you could extend your patio out from the house, and doing so would give you an area for an arbor and vines, and a place for the adults to gather. Place additional foliage here in pots, and re use the loose stone for a more tightly formed path, edged with liriope.

You aren't showing much of the raised bed, but if you keep it, look for drought tolerant full sun plants to fill it. This could be a possible good place for a bird bath.

If you run into trouble, call your local boutique nursery, and ask for a consult with their on staff designer, Take these notes, more photos than your have offered here, and a scaled drawing, and you should have the proper plants in the right amounts in no time. Best of luck.

What a cute playhouse. Have you considered rubber mulch for the play area?

When I think of kid-friendly gardens I think of what I loved to explore as a child: soft or fragrant plants, butterfly magnets, rings of tall grass I could hide in or lay within and watch the clouds, water features (even simple ones), edibles, interactive plants like snapdragons or exploding seedpods of impatiens, grass to run through, rocks or stumps arranged for hopping/tightrope walking across logs, fences that only partially exclude the view beyond, trees that are easy to climb.

Your grove of trees is lovely and has beautiful dappled sunlight - this would be beautiful planted with understory plants and perhaps a hidden bench or fort. I love the idea of leaving some grass to run, bordered/sheltered by different experiences for the kids.