Im sure /u/restricteddata will be along to provide another fantastic answer but to give you the starting points.

1. The plan had always been to drop both weapons and continue in another 3 weeks when the next trickle of bombs would begin, with a 3rd before Sept 1 and 4 in Sept, but then consideration enters in if you sue them for strategic or tactical bombings in support of the invasion.

2. The plan would change if there were obvious signs of Japanese movement to surrender, there were none.

3. Truman had approved the plan as is essentially, including the follow on attacks.

4. The 2nd bomb was not supposed to be ready till closer to the middle of August(the 12th I want to say, could have been the 11th). But bad weather forced Tibbets and the rest of the high command to push up the date to the 9th to ensure the bomber and bomb would be ok.

5. Afterwards Truman realized the scale of the situation he was in and ordered no more bombs without his direct order.

Fun fact: there is no evidence (that I have seen) from the time that Truman actually knew there was going to be a second atomic bomb so soon after the first.

We know he asked about the "schedule" of bombing on July 23rd. The reply he got, in the form of a cable, was the following:

First one of tested type should be ready at Pacific base about 6 August. Second one ready about 24 August.

Stimson wrote in his diary at the time that Truman said the above was "just what he wanted, that he was highly delighted and it gave him his cue for the warning," (the Potsdam proclamation).

You will notice this omits any discussion of the non-tested type, the Little Boy bomb, which was dropped on Hiroshima. In all of Truman's diary entries, discussions, etc., from the time, he refers only to the Fat Man bomb — the one that would have been on everyone's mind right after the Trinity test. Truman seems to have been told much about implosion and much less about the gun-type weapon. I do not think this was a deliberate deception; I think Truman had many other things on his mind other than the bomb (he was trying to jockey with Stalin).

His actions after the second bomb — immediately halting all further atomic bombing — suggest to me as well that he did not feel in control of the situation.

I take a rather dim view of Truman's knowledge of what was going on. Some things I have written along these lines:

• Why Nagasaki? — addresses the question of whether two bombs was a deliberate strategy or just because they had them ready to go

• Nagasaki: The Last Bomb — on the mishaps involved in the second bombing run

• The Kyoto misconception — on Truman's understanding of what kind of target Hiroshima was (which is relevant to the question of what Truman knew about the bombings in general)