We registered with a GP practice in Ireland that takes children, and the practice nurse was often the first person we saw for the small things. Ask whether your health centre has a nurse you can see without an appointment.
Question for parents: how did you find a paediatrician or family doctor for your children after moving? In Brazil my son saw a paediatrician directly, and the system here in Portugal works differently: the family doctor at the health centre is the first step, and I'm still waiting to be assigned one.
In the meantime, what did you do for the small things, the fevers and the ear infections that four-year-olds get every other week? Did you go private while waiting, use the emergency service, or something else? And how did you choose a doctor you trust, when you don't know anyone to ask?