Hi dr my 3yrs old baby suffering from cold cough...

: Managment 
When we talk about your baby symptoms about runny nose and vomiting we talk about viral infection , we treat it by follow your baby temparture and avoid to exceed more than 37.7 by give him paracetmol syrup and do sitz bath , then runny nose can treat by normal saline drop in baby nose to assist you to normal respiration , vomiting or diarrhea should by attention to avoid dehydration of baby by replace loosing fluid by give your by sufficent amount of fluid , should recommend there is no need for antibiotics for your baby because this is viral infection need 7-10 days  to improve completely 
: Diagnosis
Advise you to follow above instructions , but if symptoms increase as temp above 37.7 advise you to take baby to pediatric doctor to tkae detailed history , do examination and some investigations and follow your baby