How can I understand what allergies my child has?

Good question... Skin rashes are usually food or contact allergy, cough and nasal allergies are usually to airborne allergens (dust, pollen, animal dander). For food allergy, observe what seems to make it worse -- food diary is a useful tool. For airborne, the rule of thumb is, tree pollen in the spring, grass pollen in the summer, weed pollen in the fall, dust and animal dander year round but worse in winter.
Evaluation. Take your child to a board certified allergist/immunologist for evaluation and discussion.
See an allergist. . I would first discuss with your pediatrician to see if the symptoms point to allergies. They may refer you to a board certified allergist.
Observe your child. If your child has symptoms with one specific item, such as a rash when they eat strawberries, this is a good indicator. If the source of symptoms not so obvious, then talk to child's doc about when you notice symptoms - is it indoors or outdoors, during a cetain season or year round, etc. Testing can be done with either skin testing or blood tests, but it helps to narrow possibilities down first.
Observe your child. Repeated hives, diarrhea, discomfort, etc. After eating, can be from food intolerances or food allergy. Sneezing, runny nose, etc. Could be environmental allergies. Rashes can be associated with contact allergy, or eczema. Allergies to medications are possible also. By observing your child's symptoms or behaviors and discussing these with your doctor, you can see if your child has allergies.
Skin testing. The best way to understand exactly what is causing allergy aymptoms in your child is to have skin testing done by a board-certified allergist. Blood tests such as rast and immunocap are very accurate, but are only positive in highly allergic people - milder sensitivities can be missed by blood tests. Other tests you may see, such as igg tests for food allergies, have not been shown to be reliable.
See an allergist. In order to better understand whether you child even has allergies and how to best manage them, an allergist is your best bet. I would look for an allergist certified by abai or the american board of allergy and immunology. This is the "governing" body which ensures that your allergist is trained properly and maintains the most up to date knowlege.
Allergy testing. Symptoms with exposure to allergens is the first sign. Patients with allergies usually have symptoms every time they are exposed to the allergen (for example, dust mites, molds, pollens, pet danders for aeroallerens or milk, egg, wheat, soy, peanut, tree nuts, seafood for foods). If you see symptoms that occur with every exposure, then skin or blood tests can help diagnose their allergies.
An allergist. Can provide you with all your answers, depending on your child's symptoms, testing will be performed accordigly.
Allergy tests. Often doctors can tell what your child is allergic to just by the time of the year there are symptoms. For instance trees cause symptoms in the early spring; grass in later spring; ragweed in fall. Molds, dust mite and animals cause symptoms all year long. Skin testing however is the most accurate way to tell exactly what your child is allergic to. Blood tests are a less accurate alternative.