April is a fantastic time for visiting Disney World. Kids are not yet out of school, so the crowds will not be like the summer and the heat is certainly not like the summer. Most likely you will be able to wear shorts and a t-shirt during the day, but may want to bring a jacket or wear long pants if you are going to stay late as it will likely get a little cooler.
Epcot is great to visit during this time because you can see many flower displays around the park due to the Flower and Garden Festival.
I also really recommend visiting Magic Kingdom. The crowds will be less than during the summer.
Hollywood Studios is another terrific option especially if it rains because there are many inside attractions like 4D simulations to enjoy.
April 15 - The Jungle Book movie arrives in theaters.
Epcot's Flower and Garden Festival continues through May 18, 2014.