I'm sure in the next movie, we'll get an explanation of why Kirk was given command of the Enterprise, and this didn't occur right after the destruction of the Narada, because the ship suffered major damage and warp engines don't grow on trees. So, several months later, the Enterprise, under Kirk's command, starts it's first mission.
There are factors that might have contributed to Kirk getting command of the Enterprise. First off, Pike promoted Kirk to First Officer, and Spock to Captain, in his absence. The crew were mostly cadets. What else was he supposed to do? After the destruction of Vulcan, Spock's first move was to reunite with the rest of the fleet to come up with a plan. If the Enterprise had done that, most assuredly Earth would have met the same fate that Vulcan met. Consequently, Kirk's decision to go after Nero was timely and effective. After saving Pike and securing the Jellyfish, Kirk and Spock were able to defeat Nero.
Now, we have to think back to events that took place in the movie. When Nero escaped from the Klingon prison planet, he easily destroyed most (47 ships) of the Klingon Armada. They could do nothing to stop him. If you know anythings about Klingons, this would be quite a humiliating event for them. Up next, Nero goes against the Federation and he's only able to destroy seven ships. And one ship, led by a human crew was able to take Nero out. It's still humiliating for the Klingons, but it's also something they would revere. Hammering out some type of alliance between the Klingon Empire and the Federation is a much easier undertaking with the heroes who defeated Nero. Not to say that all the Klingons are going to be nice, because if the Duras were not eliminated when the armada was destroyed, there are still going to be some quite nasty Klingons around.