[The below answer comes from a friend who is a pediatrician]
You are going to find a whole multitude of answers as the field of pediatrics has many different specialities. The main groups though would be a general practitioner (you would see this doctor when you need a physical or you are sick), a specialist with a clinic (this would be doctors who see specific types of patients in clinics and hospitals, like a diabetes or heart doctor), an emergency room doctor (they see a wide-variety of patients but only in the emergency room), or an specialist who only works in the hospital (these doctors only take care of sick patients who need to be in the hospital). Each of these bigger catergories have smaller subcatergories that create a wide difference in the day-day types of children seen, the setting they are seen in, the severity of their sickness and the hours or shifts they work.
I can only answer within the "specialist in the hospital" group and say that in all honesty, each day is different. I work in a pediatric intensive care which typically takes care of the sickest patients in the hospital outside of the just-born phase. A typical day starts about 6:30 am. During this time we hear about new patients who came to intensive care unit overnight and issues current patients had. We then examine and review the status of patients who are under our care. Around 8:00 am, we go from patient to patient and review the patient's records with the residents (doctor's in-training), nurses and families. Then we make plans for the medical care of that patient for the day. This usually takes about 2 to 3 hours, depending on how complex or sick the patients are. The afternoon from about 1:00 pm to 5:00 pm deals with enacting the plan for each patient made during the morning and getting new patients into the intensive care unit and treating them. Also since patients are generally sick in the intensive care unit, the plans made in the morning might not be able to be applied later in the day, eg. a patient got sicker, and a considerable amount of time can be used in stablizing sick patients. Usually, we see about 5 new patients a day, give or take, who come to the intensive care unit. The afternoon is also mainly when all the paperwork is done. Around 5:00 pm, a team of intensive care doctors come in for the night time and we tell them about all the patients in the intensive care unit, who is sick, and what to pay attention to. Around 6:00 pm, we finish telling the night team about the patients, finish up paperwork and head home. However, like I said, each day is different. Sometimes the process feel non-stop from 6 to 6, but other times, there are breaks to sit down and take a breath, it just depends and there is no predicting it.
This is just one of the many different experiences one can have in becoming a pediatrician. I'm sure if you asked someone else, they would give you an entirely different answer, but I hope it helps some.