This is an interesting question. Did Kirk follow this "five year mission" that he narrates at the beginning of the episodes?
Space: the final frontier.
I guess it is in TOS.
These are the voyages of the starship Enterprise.
Well we do follow them around each episode.
Its five-year mission:
Kirk indirectly confirms that he was out there for five years dealing with unknowns like V'ger in TMP. Were his previous years on other ships then not as a captain or dealing with unknowns like V'ger?
to explore strange new worlds,
They do this in various episodes on planets that they appear first to discover.
to seek out new life
Yes, they discover new life (and live to tell about it).
and new civilizations,
I would suppose the First Federation, the Metrons, etc count.
to boldly go where no man has gone before.
Kirk did ignore the First Federation's warning beacon and went where they were suppose to stay out of. And past the furthest charted information in "Arena", although it was Angrily Going...
I'd say for the most part they were doing what their narrated mission statement said they would do AND doing the mundane stuff like escorting diplomats, fighting wars, policing space lanes, etc.