In the FAQ you'll find a long discussion of Kokura and why it was a preferred target. Nagasaki was added only at the last minute, and was in many ways not an ideal target — the geography was bad (the city is mountainous and that shields parts of it from the explosion, as happened in reality), it had already been bombed conventionally previously (it was not a "virgin" target), and it was less ideal from the perspective of their goal of bombing some military installation that happened to also mean destroying a city (Nagasaki had some factories produced weapons, but it wasn't the sort of place that either Kokura or Hiroshima were). This is why it was their lowest-priority target in that area. (Niigata, the other target on the list, was too far away to target once Kyoto had been removed.)

Hiroshima was considered an ideal target because it had a variety of useful characteristics: 1. it had not been bombed yet, so it was a "virgin" target; 2. it was geographically very favorable to an atomic bomb's effects (it is in a "bowl" of mountains, almost exactly the size of the blast radius of the kinds of atomic bombs they had in World War II); 3. it had a major military base in it, that would give propaganda "cover" for its being chosen as a target; 4. its many rivers made it a poor target for incendiary bombing, making the argument for using an atomic bomb on it stronger. From the perspective of the people who were making these decisions, it ticked off all of the boxes.

Tokyo was briefly considered but struck out as a possible target, at least for the first bomb(s), because it was already too bombed-out (they wanted "virgin" targets to showcase the power of the bomb), and because there were some fears that if you killed the high command, there'd be no one to surrender to you. But it is of note that there were strong advocates to bombing Tokyo with the third, un-dropped atomic bomb that would have been used if the war continued another week or so.

The Kyoto story is a long and complicated one, but it was initially the Army's most favored target (above Hiroshima) for the first atomic bomb, because of its great size and importance. It was removed from the target list by the intervention of Secretary of War Henry Stimson, who was able to get the assent from Truman on this topic. This is in fact the only serious "decision" that Truman made about the atomic bombing — everything else was him just going along with decisions other had made. But he agreed with Stimson that Hiroshima, and not Kyoto, would be the first atomic bomb target, and that Kyoto would not be targeted by conventional or atomic weaponry.

Why did Stimson not want to bomb Kyoto? He gave various reasons ranging from the moral to the cultural to the strategic, but ultimately we don't know. None of his stated reasons seem adequate to the resources he marshaled to remove it from the list. It was not a minor or easy action; it was perceived by the military as a major form of civilian interference, and they resisted it at every step, and ultimately Stimson felt he needed explicit Presidential approval to make his demand happen. I suspect ultimately there was a psychological aspect to it — Stimson was emphatically not a fan of city-bombing in general, but powerless to stop it, and had with the atomic bomb involved himself in the ultimate form of the practice. Saving Kyoto — a place he had been before, and a place that those he cared about valued highly — may have been his way of feeling he had done something to offset the destruction he had been part of.

As for the import of Truman's agreeing to remove Kyoto, I have published on this at some length — it is, I think, a pretty interesting and complex story, and one that in curious ways subverts most people's understanding of the atomic bombings and their legacy.

More can always be said, but you may be interested in /u/restricteddata's answers in our FAQ about atomic bomb targeting.