Hi Ryan!
My children, ages 5 and 10, love Character Dining Experiences. Their favorite is Chef Mickey's in the Contemporary Resort. They also love eating dinner with Chip 'n' Dale in the Garden Grill restaurant in Epcot. Your two 13 year old children may think they are too old for Character Dining, but tell them they "need to go for the 2 year old". Sometimes my oldest son doesn't want to get his picture taken with the Disney Characters and we don't make him; once he and I understood that "ground rule" he's been very happy dining anywhere; that said, he usually wants his picture taken as well!
Our family also enjoys eating in 'Ohana in the Polynesian and
Whispering Canyon Cafe in the Wilderness Lodge. Both are great family restaurants! If you go to Whispering Canyon, be sure to have one of your 13 year olds ask for ketchup! If your family likes ice cream, then I recommend stopping by Beaches and Cream at the Beach Club Resort; I think they also serve the best hamburger on property there.
In the theme parks, my children love the Crystal Palace in the Magic Kingdom and Hollywood and Vine and the
Sci-Fi Dine-In Theater in Disney's Hollywood Studios. I'm not a big fan of the food at the Sci Fi, but I have to admit the atmosphere is great and it's fun watching the movie clips (though they don't last long enough and there isn't enough of them before it repeats). Some guests also really enjoy the
50's Prime Time Cafe, but I'm personally not a fan of it. In Epcot, my children enjoy the
Coral Reef restaurant, but more for the views than the food.
Have a magical vacation!