Hi Angela! Wow it is hard to tell anyone to miss something at Walt Disney World. The problem is that most guests don't stay long enough to do it all!
There are great attractions in each park and you can see those by clicking
HERE and the reading through the six pages to see which might interest your twins. You did not mention if they were girls or boys which might make a difference in the interest level for specific attractions. Note that some will also have height requirements.
My personal suggestions for each park follows:
Magic Kingdom: anything in Fantasyland, Pirates of the Caribbean, Haunted Mansion, Jungle Cruise, and Buzz Lightyear. Don't miss the Electric Light Parade at night and also the Wishes fireworks show. If you can book the Tomorrowland Dessert Party to see that it is a great option.
Hollywood Studios: Toy Story Mania, Muppet Vision 3-D, Star Tours, and Great Movie Ride.
Epcot: Innoventions, Spaceship Earth, Turtle Talk with Crush, Journey Into Imagination, Living with the Land, Soarin, and a walk through the World Showcase. The nighttime show called Illuminations is beautiful and held at 9pm every night.
Animal Kingdom: Festival of the Lion King, Kilimanjaro Safari, Pangani Forest Trail, Rafiki's Planet Watch, Flights of Wonder, Maharajah Jungle Trek, Kali River Rapids (will get wet), and Its Tough To Be A Bug. The afternoon parade is full of characters and generally not as crowded as the parades in some of the other parks.
These are personal favorites and didn't include the roller coaster thrill rides as not sure if your six year olds are up for them.
The water parks are great too if you have the extra day and I love Blizzard Beach. Check out Downtown Disney for shopping and some entertainment. The list can go on and on so ask again specifically about recreation and dining!