A press release about the atomic bomb had been prepared well before the bomb was dropped. Shortly after a damage report confirmed that the bombing run had been successful a statement was released to the press, under Truman's name, announcing the atomic bomb. (Truman himself was still in the mid-Atlantic aboard the USS Augusta, coming back from the Potsdam Conference.) Note that in terms of time zones and the like, General Groves in Washington, DC, knew the bombings had been successful as of about 11:30pm on August 5th, Washington time. Truman's statement was released at 11am (Washington time) on August 6th. So there was a lag of about 12 hours between US officials knowing about the bombing and the public knowing. (And there was a longer-than-expected lag, several hours worth, between US officials knowing about the bombing and the Enola Gay returning to Tinian, for reasons that they never ferreted out. On the island of Tinian, the work of the Enola Gay was known by many there and they were greeted on their return with great fanfare and a large amount of celebrating.)

Shortly after the President's statement, a longer statement, with more technical information, was released under the heading of the Secretary of War, and the British also released their own statements. I can't remember if I've seen late editions of US papers reporting on the bomb on August 6th, 1945, but by August 7th it was front-page news everywhere. Several days after the Nagasaki attack, a longer, technical history of the bomb was released (the Smyth Report).

It is hard to characterize something like a "general reaction" but I find that looking at editorial cartoons from those specific days helps get a sense of it. There was a lot of what you would expect: excitement, pride, horror, hope. A lot of information that we take for granted today (e.g. what an atomic bomb really is, how it works, what it does to people, how many had died, etc.) was not immediately available. Over the course of August-October 1945, there were many mutually contradicting narratives about the atomic bombs, including those who credited them for "winning" the war (which obviously had not happened by August 6-9), and those who worried about accounts of radiation sickness from Japan (which were initially denied by the American authorities).