Rash, swelling. Classic allergic reaction are typically rashes (hives, itch, eczema), swelling of the face/lips/ etc. That occur within minutes of eating. Some may cough, wheeze, short of breath and lose color. These are common signs of allergy, esp. When baby goes from breast to bottle for first time. Vomiting & diarrhea are possible. There is a bad type of allergy that even leads to dehydration from vomiting.
Vomit/bloody stools. If your baby is frequently vomiting formula, that could be a sign of an allergic reaction. Other symptoms are blood or mucus in the stool, severe pain with feedings, poor weight gain, or eczema/severe skin rash. If there is a family history of milk allergy, that may also increase your baby's risk of having an allergy as well. Talk with your doctor if you are concerned.