1. The values of :after methods are ignored. 2. You're calling person-age from within the method so you enter infinite recursion.

What you were looking for is:

  (defmethod person-age :around ((someone person))
    (let* ((age (call-next-method))
	   (category (if (< age 16) 
		         :child
		         (if (>= age 66)
			     :senior
			     :adult))))
       (values age category)))