My question is: Is there an official reason why the Enterprise went on their 5-year Mission? Or was it just to explore new worlds?
The obvious answer from the intro is:
Space: The final frontier. These are the voyages of the Starship, Enterprise. Its 5 year mission: To explore strange new worlds. To seek out new life and new civilizations. To boldly go where no man has gone before.
That said, the Captain's logs add flavor and texture to the mission purpose throughout the series.