At what age does a child stop seeing a pediatrician and start seeing an adult medicine doctor?

Various. Many children as they get older see the pediatrician as an indication of childhood and want to see an adult doctor as proof of their maturity. There are some adolescent doctor specialists as well. Some children with disease needing frequent visions will stick to their pediatrician for a long time as they know each other well. So there are various answers and no absolute time.
Transition. That depends on the person, the resources available and on any underlying problems. For most places it is 18 or graduated high school. Some kids with chronic illnesses will see their pediatrician or pediatric specialists until 21 or longer. Some teens get tired of seeing the "baby" doctor and want to go to adult med doctor sooner. As long as someone is comfortable managing care it's up you.