Many people have a memory of the moment they realized that Barack Obama was likely to become president. For some, the moment came during the primaries, perhaps when they heard the speech he gave after the New Hampshire primary.
Others, and I know several who were present, will point to his announcement in Springfield, Illinois that he was running for the presidency.
There are others who will go back even further and point to his Keynote address at the 2004 Democratic Convention.
My moment was a bit more obscure, but it had to do with libraries. I recall becoming convinced that he would run for president, and would likely win. I expected it in 2012 or 2016 (he was early in his first-term as a U.S. senator, remember), but the past two years has been eerily like what I anticipated that day.
In June 2005, Barack Obama spoke at the American Library Association conference in Chicago, Illinois. I can’t find a video of the speech, but the transcript is here. I did find a discussion of his favorite book, however.