The Enola Gay crew were all members of the 509th Composite Group, a special group put together specifically to be trained for the delivery of atomic bombs. They required training on special B-29s that had been modified as part of Project Silverplate to accommodate the weapons (which included removing most of their armor and armaments), and they had to be familiar with the non-standard ballistics of the weapon shapes (which were much heavier and more compact than traditional bombs).

Most members of the group were not told about their special cargo, though. They practiced with weapons known as "pumpkins," basically atomic bomb casings filled in with high explosives — big bombs by pre-atomic considerations (six or seven tons of TNT equivalent). They also practiced the special delivery maneuvers they'd need to use; e.g., dropping their bomb then making a quick 155º diving turn so that they would be many miles away from it once it went off. This was not a standard thing to attempt with a B-29, and it was a difficult maneuver (a postwar B-29 attempting it in training did crash at one point).

The total 509th, including all support personnel, was some 1,500 people. It was a big organization. The head of it was Col. Paul Tibbets, who was the pilot on the Enola Gay mission. Tibbets knew the whole story about the bombs, but the rest were mostly under the policy of "compartmentalization," or "need to know."

But many of them had an inkling. Atomic bombs were something they had heard of as a possibility, in popular culture. And they knew that their big, "special" bombs were probably more than just large HE bombs. They also flew several "pumpkin" missions over Japan, to practice for the actual bomb, once they relocated to Tinian. The fact that they were different and "special" were known to all involved — as was the fact that they had some kind of secret mission that even they didn't know about. A poem was written about them prior to the atomic bombings:

Into the air their secret rose.

Where they are going nobody knows.

Tomorrow perhaps they’ll return again and we’ll never know where they’ve been.

But take it from one who knows the score; the 509th is winning the war.

But the crew of the Enola Gay was not informed of the specifics until August 4th, 1945, when they received their strike briefing on Tinian. Tibbets told them, at that time, about the size of the blast they were to be expecting, and showed the crew pictures from the Trinity test. It wasn't mentioned that they were atomic bombs, but many of the crew apparently suspected as much.

When they returned back to Tinian, they were greeted with cheers and photographers. Soon the secret would be "out."

As for the crews' thoughts — again, they didn't know much before, so no concerns were voiced. (When they heard they were going to be dropping a big bomb, and doing so without any fighter escort, they were a little apprehensive, but I don't think that was about the morality of the matter.) Afterwards, as far as I know the crew was very supportive of the bombing. And why not? They believed that they ended the war. They were told they were great heroes. They went from having an ambiguous and mysterious wartime assignment to being part of one of the most gripping moments in history. Tibbets in particular went to his grave one of the strongest proponents for the atomic bombings as there were.

They did sometimes voice some, tempered regrets about the casualties. Morris Jeppson, the assistant weaponeer on the Enola Gay, later said that he regretted "that people were enveloped in that destruction" that he could see from the air. Another one of the crew called the view a "peep into Hell." One of the crew wrote in his logbook:

I am certain the entire crew felt this experience was more than anyone human had ever thought possible. It seems impossible to comprehend. Just how many Japs did we kill? I honestly have the feeling of groping for words to explain this or I might say My God what have we done. If I live a hundred years I'll never quite get these few minutes out of my mind. Look at Capt. Parsons, why he is as confounded as the rest and he was supposed to have known everything and expected this much to happen.

(Emphasis in the original.) The same crewman later wrote about this:

People get the wrong meaning, that we immediately felt sorry. This was not the intent. The intent was that it was so enormous — human beings developed something to destroy a whole city at a time — it was utterly incomprehensible.

Which is to say, it is a complex, mixed emotion. To be taken aback by the destruction is not to say you think it should not have happened, or that they regretted being part of it. Again, the atomic bombings are a complex topic even today, but especially for a US service member in World War II.

On the atomic bombings, Rhodes' The Making of the Atomic Bomb is still a classic. John Coster-Mullen's Atom Bombs is a wonderful (although self-published) resource, especially as Coster-Mullen did quite a lot of interviews with crewmembers and compiled many other interviews that had been done by others.