Why should kids do chores? To learn to take care of themselves - the goal of parenting, after all, is to teach independence - and to teach a child how and why to contribute to the family as a whole. Furthermore, completed chores give a child a sense of accomplishment, just as learning any new skill.
The way to teach a child his chores, at any age, is to tell him about the chore, show him how to do the chore, watch while he does it - maybe several or many times, and then let him do it alone. You may need to remind him how or when; you may even have to do this until he's 17, but it will eventually get easier.
As soon as a child can walk, teach him to place toys in a basket or box when he's finished; if you take this one step further, you can train him to remove a new toy only when the first is put away. No, he will not do it perfectly and no, he will not do it every time. But if you start early, the stage is set and the child will treat this clean-up as a matter of course.
When a child turns two, depending on his height of course, he can place napkins on the dinner table, or put away spoons and plastic cups or containers after the dishes are washed. He can wipe up a spill with a paper towel and match up his socks. Two-year-olds love to help - he can use a small broom and dustpan, or fold towels. No, he will not do these things perfectly.
Three-year-olds can make their beds and learn to fold their laundry; they can help set the table. If you have pets, a three-year-old can also be taught to feed them and keep their water dish filled. At four, a child can be trusted to carry non-breakable or sturdy dishes to and from the table and, if he can reach, can even be taught to fill and start the washing machine, and move clothes to the dryer.
As a child grows in understanding, height, and strength, he can and should be taught to care for all of his own possessions, his bedding, his clothing, and yes, even the actual dusting and vacuuming of his room. An eight-year-old is perfectly capable of doing all these things.
The right age to do yardwork can be difficult to judge - there is a much greater safety factor in using powerful and dangerous equipment such as mowers and clippers. Rakes, however, are fairly safe when used appropriately, and no one ever died from putting leaves in a large bag; sweeping can be done by the smallest child, as can pulling weeds.
A ten-year-old may be able to use a push mower, after a talk about safety and under close supervision; a twelve-year-old should be able to handle trimming, again, under supervision. Teenagers, of course, are able to do all sorts of chores; there may, however, be a small skirmish when the issue of pay arises.
A family should be a cohesive unit in which all members contribute to the well-being of the whole. There's an old expression, Many hands make light work, which is good advice to follow; when all play a part in daily life and daily living, all can take pride in the family unit. And when your children are grown and on their own, they will be truly independent.