My kids love Tangierine Cafe in the Morocco Pavilion (they love the shawarma platters with tabbouleh and baklava, but there's a children's menu with burgers and such for your picky eater). We've also had unexpectedly good food at the Liberty Inn (U.S. Pavilion), which has a huge indoor dining area.
For table service, they love the Coral Reef (but only with a table by the tank - we let the receptionist know we're happy to wait as long as necessary for one to open up) and Biergarten (delicious buffet with plenty of options for picky eaters, plus the live oom-pah band is a big hit, and they'll invite people to get up and dance, which the kids enjoy).