We keep all of the children's plastic cups, bowls and plates in a bottom cupboard next to the dishwasher. (Side note: This is the only cupboard that the toddlers and babies are allowed into. They can play to their heart's content, but it's usually in a frightening state of disarray.)
We keep all the other plates and bowls on a buffet that we purchased from IKEA. An unused dresser would work just as well. This is right next to the dining room table to make table setting easy. The silverware is kept in restaurant-style containers (also from IKEA) on the buffet.
The children start setting the table at around 4 years-old with some help. They start unloading the dishwasher at around 6 years. They unload all but the glasses and glass containers which are too high for them to reach.
Here is our buffet complete with our books and Bibles ready for Cocoa Time!