I was there with my family as a 14 year old.
It was an east coast Disneyland for us. (Being so many of the exhibits put together by Disney.)
I mostly remember the Ford Exhibit, where you rode around it in Ford convertibles, the Lincoln audioanimatronic exhibit, The GM Pavilion, GE’s Progressland, and eating in the Wisconsin Pavilion with the world’s largest (17 ton) cheese and a cafeteria cheap enough for our family of 6 to all get steak dinners.
It was also a hoot seeing Shea Stadium on the way out.
Here’s a pretty good site where you can click on the map and see each exhibit that was there.
Am I the only person who saw this thread title and immediately thought of the song “Ana Ng,” by They Might Be Giants?
All alone at the '64 World’s Fair
80 dolls yelling “small girl after all”
Sitting at the DuPont Pavilion,
Why was the bench still warm? Who had been there?
I went too. But since I was less than 1 year old at the time, I don’t really remember it. My Dad tells me I really, really enjoyed the colored lights.
I know that my father went in early 1965 when he was seven.
Sheesh! Would someone check my pulse to see if I’m still alive?
RTF At least I didn’t see Teddy hit a homerun at that event. 
I went three times but, being the same age as Biggirl, was also at the “Lookee at the colored lights!” stage. The pictures are cute–folks seemed to dress up a little to go there, and apparently there were only two haircuts available to the male population of the US that year. As a Bronxite, I of course detest Robert Moses, but back then in my innocence I enjoyed the Fair without brooding about the chairman.
In the 70’s, before things were cleaned up, we used to go into the NYS Pavilion and run around the state map, with the detaching spines of the Pavilion’s roof hanging treacherously above us. It’s a lot nicer around there now.
This guy has a site with pictures of the ruins of the Fairs in Flushing: http://www.modern-ruins.com/fair/fairphotos.html