Each captain only gets a handful of missions but they're held together by the mystery of this Vulcan scientist. In trying to help her, he uncovers an elusive mystery that flows through to the eras of Kirk and Picard. Without giving too much away, the story starts with Captain Archer's attempts to help a female Vulcan scientist named T'Uerell. Given that DC Fontana and Derek Chester are responsible for the story, it should please Trek fans. You'll definitely have to draw some of your own connections between the eras if you want to follow the thread from Archer's first encounter to Picard's final showdown but it holds up well enough to provide reasonable continuity and framework for the game.
It's tough to find a single story that ties together the original series, Next Generation and Enterprise (and throws in some references to Deep Space Nine and Voyager) but the game's writers have managed to pull it off quite well.