They can be combined, vomiting with wheezing, or hives with wheezing. If there's a combination of two or more, or one symptom that's life-threatening, that's called an anaphylactic reaction, you should treat it right away with the EpiPen and Benadryl, and follow with a visit to the emergency room. There are two ways, depending on the age of the child. Both have disadvantages. Skin tests have false positives.
Blood tests are invasive, and are only as good as the lab that does the testing.
Many doctors do both. But interpreting it is an art, rather than a science. Parents need to keep an accurate food diary for two weeks, and that's when you can pick up really useful information. Before testing, most allergists recommend eliminating the suspected food for two weeks. You need to learn how to read a label, so you can completely eliminate the suspected food.