I have gone to Disney World solo 4 times so far (with a 5th planned for later this year). Some of my favorite things to do are:
Getting to the Magic Kingdom early, then just strolling down Main Street taking in the sites, listening to the soft music playing in the background, and looking at the windows in the upper floors of the buildings to see what they all say. Also, I like grabbing a snack over at the Sleepy Hollow (their ice cream cookie sandwiches are excellent!) and grabbing a table near the wall. Its quiet and has a beautiful view of the castle and its reflection in the water. I also like grabbing a seat (early) on the 2nd floor of the train station to view SpectroMagic & Wishes. The Crystal Palace is my favorite place to dine at this park.
Wandering around EPCOT and taking the time to soak everything in. I especially like Future World at night after its closed. Very quiet, nice soft music and the buildings are lit really nice, especially the Imagination pavilion and Spaceship Earth. I usually try to get a PS at the Rose & Crown before Illuminations, then watch them from my table, while enjoying a great meal. The Rose & Crown and Le Cellier are my favorite places to dine at this park.
At Animal Kingdom, I like to take my time and walk all the way around the Tree of Lift. Everyone goes to the front to take pictures, then to Its Tough, but hardly anyone walks all the way around the tree. Theres a waterfall and it is just wonderful to see all the different animals carved into the tree while there is soft music playing in the background. Tusker House is my favorite place to dine at this park.
At MGM, just wandering around. I like to get to Fantasmic early to get a great seat. 50s Primetime is my favorite place to dine at this park.
I always have a book with me, so it helps to pass the time waiting for a parade or show to start.
I also like treating myself to a decadent sundae at Ghiradellis at DTD. YUMM!!!!!