Ascending trees, small hills and climbing walls are great forms of exercise and fun. Doing it without footwear can provide better traction and allows for a better feeling of the surface and how to grip it. Bare toes can help with grip much better than shoes and allow quick adjustments in position as conditions change.
As always with climbing, loose surfaces need to be watched for as well as sharp or jagged rock. After practice, climbing barefoot can add new sensations and enjoyment of different textures.
Outdoor play areas with climbing walls, or at pools, are the easiest places to start. Indoor climbing courses may be more difficult to allow for bare feet. If a business will not allow climbing to be done barefoot, and no other venues are available, then minimalist footwear is the next best option.