One thing I want to say up front: there are many legitimate interpretations of the atomic bombings. Were they a good thing, or a bad thing? Were they moral acts, or essentially war crimes? Were they necessary, or not? Were they avoidable, or were they inevitable, once the US had the weapons? What would the most likely scenario have been if they weren’t used? How should we think about their legacies? And so on, and so on. I’m not saying you have to subscribe to any one answer to those. However, a lot of people are essentially forced into one answer or another by bad historical takes, including bad historical takes that are systematically taught in US schools. There’s lot of room for disagreement, but let’s make sure we’re all on the same page about the broad historical facts, first. It is totally possible to agree with all of the below and think the atomic bombings were justified, and it’s totally possible to take the exact opposite position.
There was no “decision to use bomb”
The biggest and most important thing that one ought to know is that there was no “decision to use the atomic bomb” in the sense that the phrase implies. Truman did not weigh the advantages and disadvantages of using the atomic bomb, nor did he see it as a choice between invasion or bombing. This particular “decision” narrative, in which Truman unilaterally decides that the bombing was the lesser of two evils, is a postwar fabrication, developed by the people who used the atomic bomb (notably General Groves and Secretary of War Stimson, but encouraged by Truman himself later) as a way of rationalizing and justifying the bombings in the face of growing unease and criticism about them.
What did happen was far more complicated, multifaceted, and at times chaotic — like most real history. The idea that the bomb would be used was assumed by nearly everyone who was involved in its production at a high level, which did not include Truman (who was excluded until after Roosevelt’s death). There were a few voices against its use, but there were far more people who assumed that it was built to be used. There were many reasons why people wanted it to be used, including ending the war as soon as possible, and very few reasons not to use it. Saving Japanese lives was just not a goal — it was never an elaborate moral calculus of that sort. Rather than one big “decision,” the atomic bombings were the product of a multitude of many smaller decisions and assumptions that stretched back into late 1942, when the Manhattan Project really got started.