Richard,
There are a lot of great places to eat at both, but clearly with kids (mine are 5 & 7), something more family friendly is important. Here are a few choices at each;
Epcot
When we travel to Walt Disney World,
Via Napoli in the Italy Pavilion has become one of the few restaurants we HAVE to visit every time. The pizza is great for both finicky kids, and parents wanting something a bit out of the ordinary. And there are other safe choices, like pasta.
Another Epcot choice could be
Chefs de France. They have some wonderful French cuisine, but also have the ability to make foods closer to American standards for kids.
Downtown Disney
Most people run to recommend both T-REX and the Rainforest Cafe. But they are not my favorites. I would instead opt for
Wolfgang Puck Grand Cafe, for both higher end cuisine, and family friendly options.
My second choice may seem odd, but I would go to the
Earl of Sandwich. I assumed you were looking for table service, but with this quick service choice you get a great meal, and you get back to shopping faster! Pair that with a sweet treat from
Ghiradelli, and you are set for the night!
Write me back with any follow-up questions!
Have a magical (and delicious) vacation!