See below. A lot of the diagnosis can be based on the history of your child's problems, and the exam. Then specific testing can be through blood work or skin testing.
History, exam, tests. The diagnosis of allergies is achieved after a detailed history of symptoms and timing of symptoms, a thorough physical exam and allergy testing. The preferred testing is skin testing which uses plastic applicator and not needles with an answer in 20 minutes. This is best accomplished by an allergist trained in testing and interpreting tests in young children.