At what age does a person stop growing height, and also shoe size?

Around 18: Most of the bones of the foot stop growing around the age of 18. By looking at an x-ray of the foot a doctor can tell if the bone has stopped growing. Some people may continue to grow a short time after 18 but that is the age for most people.

Skeletal Maturity: A person stops growing when the growth plates in the bones close, which is medically known as "skeletal maturity." typically this is 16-18 for girls and 18-20 for boys. Bones in the foot generally mature about 2 years sooner than the long bones that affect height. Keep in mind that these are averages, so some people are on the very high or low ends of the spectrum.