Trek III is the film that expanded the universe the most, and it has the grandest scope. So many new ships. So many new characters. Movie Klingons for the first time. It's also the most daring. Killing David, destroying the Enterprise, turning the crew into renegades. And James Horner's soundtrack! The soundtrack!
Definitely! Very influential on the rest of the franchise, moreso than any other Trek film before or since. TSFS...
1. Introduced the notion to Trek that the hero's ship is fair game for destruction during an adventure, but don't worry, it'll be replaced in the next film (or next episode, in the final season of Deep Space Nine).
2. Was the debut of the Excelsior, which represented 90% of Starfleet vessels in the TNG era.
3. Began the trend of the Klingons being the most over-used alien race in Trek lore.
4. Unveiled the Klingon Bird of Prey, which became at least as common in the Klingon space armada as the Excelsior did in Starfleet's, in multiple sizes, even! Also, the Romulans were no longer the alien race with the cloaking attack vessels.
5. Featured that-there Shroomdock, showing use in no uncertain terms that Starfleet's main base was situated in orbit around Earth.
6. Had the heroes no only disobeying orders from Starfleet Command, but also becoming outlaws. (Don't worry though everything will turn out fine, and their illegal activities will have saved the day!)