Year Event Relevant Quotes Annotations
2233 James T. Kirk is born. TOS The Deadly Years

KIRK
I'm 34 years old.

Assuming a 2267 date for TOS The Deadly Years
2242 Gary Mitchell is born. TOS Where No Man Has Gone Before

A computer screen states Mitchell's age as 23.

Assuming a 2265 date for TOS WNMHGB
~2250 Kirk and Mitchell meet for the first time.

TOS Where No Man Has Gone Before

KIRK
Our task: Transport down a man I've known for 15 years, and if we're successful, maroon him there.

[...]

MITCHELL
Hey, man, I remember you back at the academy. A stack of books with legs. The first thing I ever heard from upperclassmen was, ''Watch out for Lieutenant Kirk. in his class, you either think or sink.''

There are two possibilities:

1.) Kirk was a Lt. at age 17 and Mitchell attended the Academy at age 8. At first glance that seems highly unlikely but don't forget that Star Trek was supposed to be "Hornblower in Space". In the sailing age, underage midshipmen were the rule, rather than the exception. Admiral Farragut for example was 9 years old when he joined the US Navy. There's also a line in the TWOK script stating that Midshipman 1c Peter Preston was 14 years old and a cut line from DS9 Resurrection states that a 17 years old Worf served as an ensign about the USS Hawk. But that theory is irreconcilable with TOS Shore Leave, so let's forget about it.

2.) Kirk and Mitchell knew each other before Mitchell attended the Academy.

Problems: none

~2250 Kirk's first year at Starfleet Academy. One of his instructors is Benjamin Finney.

TOS Shore Leave

FINNEGAN
I never answer questions from plebes, Jimmy boy.

KIRK
I'm not a plebe. This is today, 15 years later

TOS Court Martial

STONE
Let us begin with your relationship with Commander Finney. You knew him for a long time, didn't you?

KIRK
Yes. He was an instructor at the academy when I was a midshipman.

A plebe is usually a first-year student.
~2254 Ensign Kirk graduates from the Academy and is stationed on the USS Republic under the command of Captain Garrovick.  

STONE
But that didn't stand in the way of our beginning a close friendship.

KIRK
His daughter Jame, who was here last night, was named after me.

STONE
It's common knowledge that something happened to your friendship.

KIRK
It's no secret. We were assigned to the same ship some years later. I relieved him on watch once and found a circuit open to the atomic matter piles that should've been closed. Another five minutes, it could have blown up the ship.

COMPUTER
Ship nomenclature-- specify.

KIRK
United Starship Republic, number 1371 .

[...]

SHAW
With reference to Records Officer Finney, was there in his service record a reported disciplinary action for failure to close a circuit?

PERSONNEL OFFICER
Yes, ma'am.

SHAW
Was the charge in that instance based upon a log entry
by the officer who relieved him?

PERSONNEL OFFICER
Yes, ma'am.

SHAW
And who was that officer?

PERSONNEL OFFICER
Ensign James T. Kirk.

TOS Obsession

McCOY
Captain Garrovick was very important to you, wasn't he,Jim?

KIRK
Yes. He was my commanding officer from the day I left the academy.

The ST Chronology suggests that Kirk served as an ensign aboard the Republic while still attending the Academy, but the dialogue doesn't really support this theory.
2254 Garrovick takes command of USS Farragut and invites Kirk to join his crew. The Farragut is Kirk's first deep-space assignment and he is promoted to Lt. (JG). One of the Farragut's first missions is a survey of Tyree's planet. TOS A Private Little War

KIRK
So it seemed to brash young Lieutenant Kirk on his first planet survey.

[...]

When I left there 13 years ago, the villagers had barely learned to forge iron.

TOS Obsession

SPOCK
Among the survivors was a young officer on his first deep-space assignment: James T. Kirk.

Assuming a 2267 date for TOS A Private Little War.
2257 Lt. Kirk's encounter with the Dikironium cloud. TOS Obsession

McCOY
I was speaking of Lieutenant James T. Kirk of the starship Farragut. 11 years ago, you were the young officer
at the phaser station when something attacked.

Assuming a 2268 date for TOS Obsession
~2257 Mitchell's first year at Starfleet Academy. Lt. Kirk is an instructor at the Academy.   Earliest possible date.
2265 Capt. Kirk takes command of USS Enterprise   This leaves awfully little time for promotions to Lt. Cmdr. and Commander. On the other hand, Geordi La Forge advanced from Ensign to Lt. Commander within 3 years.