Spock Prime advised Kirk to start a defining friendship with young Spock.
Spock started to develop a respect for Kirk when he saw Kirk's "won't back down" determination to take on the Rommies, save Earth, and retrieve Pike. Kirk started respecting Spock when Spock had finally shown a little more flexibility than logic would've demanded, and his stricture to duty when Spock returned to the bridge to help them plan Kirk's attack.
Fascinated they got as far as they did, Spock shows even more respect for Kirk, and actually refers to him as "Jim" when he cites the odds for the success of their mission as they prepare to part ways....Kirk to find Pike, Spock to fly Spock Prime's ship. Spock is even more surprised by Kirk's blind confidence in him.
Seeing Kirk and Spock stand shoulder to shoulder as Nero continues to make hostile gestures despite his ship's impending fate. And when Kirk and Spock have a crossing of the minds, if you will. Kirk invokes logic to save Nero and his crew, and Spock shows that he prefers a little "illogic" in this case. Both seem to impress the other. When Nero finishes his posturing, Kirk and Spock casually walk back to their duty stations as Kirk orders Sulu to "fire everything they've got".
After they escape the shockwave, everyone's having a laugh of relief, and Kirk and Spock nod to each other in mutual acknowledgment that they both managed to pretty much pull off the impossible.
Spock Prime advised young Spock to start a defining friendship with Kirk, citing the things they could accomplish together as friends, as opposed to being adversaries. Why should Spock question his own logic, even if it came from his future self?
And when Spock reported to Captain Kirk for duty, and submitting his candidacy for "first officer", Kirk replies "It would be my honor, Commander."
You could see all this as the start of a great friendship that will define Kirk and Spock, and ultimately define Kirk, Spock, and McCoy.