Although I really didn't know who she was at the time (I was 7), I remember watching Americas Most Wanted when they broke in to announce Princess Diana's death.
On 9/11, I was about 20 miles from the Pentagon. We had just gotten to school and our teacher took us to the school library to start working on a project. I passed by the librarian's office while looking for a book and happened to see they had the TV on the Today Show, where I saw that the Twin Towers were on fire, but I didn't think much of it and went on about my day. Later that afternoon, our teacher told us we weren't going to go out for recess because it was her birthday (which was true, she mentioned it the day before), again thinking nothing of it, we all went on about our day.
I got on the bus that afternoon and turned my Walkman radio on DC101, like always, which is when I heard the actual news of what happened. We weren't told about it in class, because one of my classmates had a parent who worked at the Pentagon and the school was trying to confirm if the parent was safe or not, luckily he was safe as he didn't go to work that day for whatever reason.