When I was a kid, watching first run Star Trek
(and reading everything anywhere I could find about it), the impression I got was that the Enterprise
and her sister ships were the biggest and best in the fleet, Earth's first true deep-space explorers, and that her mission was exactly what was stated at the start of every episode, "to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man has gone before."
That belief worked well enough during those first episodes, when the Enterprise
seemed to be constantly encountering new, unknown alien races, and encountering other humans only at distant frontier outposts like those of Robert Crater and Roger Korby.
Somewhere along the line, I learned that the Enterprise
was one of only twelve of those biggest and best deep-space explorers, and it wasn't hard to imagine they were all pretty much purpose-built to be provisioned for five years and sent out to explore "where no man has gone before," just like the Enterprise
Of course, that all changed once the scripts starting calling for frequent interactions with well-established colonies, Federation member worlds, various Federation bureaucrats, and the occasional traveling theater group. But I really liked the feel of those earlier "out on the frontier" episodes.