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In 2254, James T. Kirk served as an ensign aboard the Republic, along with his friend Ben Finney. During a duty shift, Finney accidentally left a circuit open to the atomic matter piles which could have resulted in the destruction of the ship; Kirk logged the incident, and Finney was denied promotion. (TOS: "Court Martial")
By the early 24th century the Republic was assigned as a training ship for Starfleet Academy cadets. Although it would continue to serve the fleet through as late as 2374, it had not left the Sol system since the 2320s. (DS9: "Valiant")
In "Obsession" Kirk says that Captain Garrovick was his commanding officer from the day he left the Starfleet Academy till his death in 2257. This would suggest that Garrovick was the CO of the Republic in 2254.
Although many reference sources included the Republic as a Constitution-class heavy cruiser, its status was uncertain. Theories about the Republic began early on with reference works published in the 1960s. Actually, these theories were somewhat backed up by production sources as the Star Trek: The Original Series producers composed a list of fourteen ships at the start of its second season, belonging to the Constitution-class, then still referred to as "Starship-class" by them and including the Republic. (The Making of Star Trek, p. 165) However, the ship was never shown on screen and its configuration was never canonically established.
The Republic's registry number was derived from the "Operation: Retrieve" star charts used in scenes cut from the The Undiscovered Country as released , whose creator, Michael Okuda, propagated the number Gregory Jein had postulated in his influential "The Case of Jonathan Doe Starship" article, published in the April 1973 issue 27 of the T-Negative fanzine.  But unlike many of his other conjectural Constitution-class registry numbers, the 2006 remastered version of the series afforded no opportunity to retcon conjecture to canon.
Ronald D. Moore, the writer of "Valiant", stated that the mention of the Republic in that episode was probably the same ship that Kirk served on. (Star Trek Encyclopedia 3rd ed., p. 653) It was debatable whether a ship that old would still be useful as a training ship over 125 years later, except possibly as a study of antiquities. However, several 21st century nautical services from a wide variety of seafaring nations continue to use sailing ships as training vessels (the US Coast Guard, for example, has used the barque Eagle for cadet training since the 1940s). In TNG: "Relics", Geordi La Forge mentioned that many systems are essentially unchanged since Montgomery Scott's time, if this was so, a ship as old as USS Republic, especially if regularly refitted and upgraded, would be just as useful as a brand new ship in that regard.
According to a ship listing included in the plans, and not seen in the extended cut of the sixth movie, the Republic was commanded by H. Zimmerman, and this ship was located in Sector 22956. The ship's captain was named after Production Designer Herman F. Zimmerman. 
An early comic book from Marvel's Star Trek series depicted this as a Baton Rouge-class (β) vessel, based on the illustration of that class by Rick Sternbach in the Spaceflight Chronology reference (and described in John M. Ford's novel "The Final Reflection"). This would make sense for the registry number and the general era the ship existed during; however, it bears no regard for the creator and producer intentions, nor was it confirmed by any filmed Star Trek, since these licensed books and materials are non-canon.
At the beginning of the video game Star Trek: 25th Anniversary, the Enterprise engaged the Constitution-class starship Republic in a mock battle. In the game, the commander of the Republic, Captain Richard Cornelius Patterson, sends you his congratulations if you defeat his ship in the exercise. If you lose the mock battle, he sends the message "Better luck next time..." Later in the game, the Enterprise finds the Republic adrift, wrecked by a duplicate version of the Enterprise, her crew slaughtered and the ship in a decaying orbit.
In the Star Fleet Battles game series, the USS Republic was near a convoy on the first day of the Romulan entry in the General War and successfully fought off five Romulan ships with the help of a police ship and the convoy itself. The story and game scenario were entitled "And To The Republic" and the scenario was also rewritten for the Federation Commander game system.