A few quick tips for working with kids (especially the toddler & school age kids, where non-pediatric-minded people seem to struggle most):
1. Put yourself at their eye level as much as possible. Most kids are at least a little afraid of the doctor, and looming over them by 2-3 feet is not going to assuage that fear. You can squat, kneel, or sit, depending on the situation.
2. Talk directly to the kids, using age-appropriate vocabulary. Kids don't like being ignored, and appreciate being included in their own care to some extent.
3. Discuss something non-medical to open. Comment on their shoes, or their name, or their hairstyle. For school-age kids in clinic, you can simply ask them about their day.
4. If your institution allows it, ditch the white coat. Seriously, I finish residency in a few months, and my long white coats are still sitting in plastic in a closet. I've never worn them. You, too, will survive.
5. For some toddlers, it helps to demonstrate exam techniques on parents or yourself to make them feel less invasive. Sometimes for the really young ones it's enough to hand them the working end of your stethoscope so they realize it's not going to hurt them. For a 3 or 4 year old, enlist them to help put the diaphragm in the right spots.
6. Accept that no matter how good you are with kids, you can't win them all. I usually do pretty well, but on a busy day in clinic I usually have at least one child that bursts into tears the moment we make eye contact.