As soon as your child can begin any ball and eye / hand and feet coordination, he or she is ready to start playing tennis!
Some programs begin as early as age 3; I begin group classes at age 5. Ages 5 through 7 require learning more Agility Balance Coordination and Speed and have a shorter attention span, while ages 8 to 10 have better balance and are able to follow instructions. When we begin tennis, I start them at same age group levels, and we progress (ROGY = Red, Orange, Green, Yellow Ball) from a 36 ft court to a 60 ft. court to a 78 ft. court, doing singles and doubles.
Your child will move up to the next level as they attain the necessary skills. My goal is to get them having fun, be excited about coming to tennis, getting fit and doing practice homework between classes. As they pass age 10 they all have different goals. Then we, the coach and the parent, must be there to motivate and praise the efforts and keeping a schedule a routine and help them follow their goal, as what they learn on the court gets them ready for life.
They'll learn etiquette, sportsmanship, math, cooperative vs. competitive, following instructions, being a team player, being a leader, knowing their score and opponents -- tennis is great, and there is always the next level...