In 2254, James T. Kirk served as an ensign aboard the Republic, along with Benjamin Finney, a friend of Kirk's. 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 continued 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 until his death in 2257 while on the USS Farragut, suggesting that Garrovick may have captained the Republic. However, multiple mentions in other episodes date Kirk's Academy years from 2252 to 2257. This suggests that Kirk was assigned to Republic as an ensign while he was still in the Academy. This is also consistent with him saying he was a "young lieutenant" when he visited Neural in 2255. (TOS: "A Private Little War") Garrovick, therefore, may not have been Republic's captain.
Although many reference sources included Republic as a Constitution-class heavy cruiser, its status in canon is, to date, uncertain. Kirk himself stated in the episode "Court Martial" the ship's nomenclature as "United Starship Republic, number 1371". Theories about Republic began early on, with reference works published in the 1960s. Actually, these theories were backed up by production sources such as Star Trek: The Original Series producers D.C. Fontana and Robert Justman, who composed a fourteen ship list at the start of its 1967 second season, belonging to the Constitution-class, then still referred to as Starship-class by them and including Republic. (The Making of Star Trek, p. 165) When he embarked on his influential "The Case of Jonathan Doe Starship" article and became aware of the original reference, TOS fan Greg Jein, like many other contemporary fans, proceeded from the assumption that Republic was no longer in service as a "front line" ship, as its registry number was not included on a starship status chart in Commodore Stone's office at Starbase 11 in "Court Martial". However, Jein has included the ship in an effort he made to marry Constitution-class vessels to established registry numberss, citing Republic as an older "Mark VI" sub-class variant. Yet, as the ship was never shown on screen (nor did the remastered version of the series allow for retconning the ship into any of the episodes, as was the case with USS Intrepid), either as the ship proper or in any graphical representation, its configuration was never established in canon.
Republic's registry number is, as above indicated, derived from "Court Martial" itself, where Kirk explicitly states the number in dialogue. Senior Scenic Artist Michael Okuda – who is, like Greg Jein, a TOS fan – was also aware of the original reference and used the correct registry in his below-mentioned "Operation Retrieve" mission charts, and thereby overturning Jein's assumption. Through reference books he subsequently wrote and following the original intent of the TOS producers, Okuda has propagated the notion that Republic belonged to the Constitution-class. (Star Trek Encyclopedia (4th ed., vol. 1, p. 158; vol. 2, p. 217) This notion has been adhered to in every (in-universe) reference work since, encompassing the Star Trek Fact Files and the more recent Haynes Enterprise Manual.
Ronald D. Moore, also an avid TOS fan and the writer of "Valiant", stated that the mention of Republic in that episode was probably to the same ship that Kirk served on. (Star Trek Encyclopedia (4th ed., vol. 2, p. 217) 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 observe the practice of continuing 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, Republic was commanded by H. Zimmerman, and this ship was located in Sector 22956. The ship's captain was named after Production Designer Herman 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 book (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.
"Who's Who in Star Trek 1" gave its registry as NCC-1373.
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 Republic, Captain Richard Patterson (β), sends the player his congratulations if Republic was defeated in the exercise. If the player lost the mock battle, he sent the message "Better luck next time…" Later in the game, Enterprise found Republic adrift, wrecked by a duplicate version of 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.