Gene Roddenberry’s first try, dated 8’1’1966:

This is the story of the United Space Ship Enterprise. Assigned a five-year patrol of our galaxy, the giant starship visits Earth colonies, regulates commerce, and explores strange new worlds and civilizations. These are its voyages … and its adventures.

I think that’s terrible and, trust me, his next draft was worse. The intro emerged thru collaboration. For 10 days Roddenberry traded drafts with producer Robert D. F. Black and associate producer Robert Justman. Then he wrote out the final and Justman ran it over for Bill Shatner to record. All this was very last minute.
From the memos reproduced by Inside Star Trek: The Real Story, we can see that Black brought in “the final frontier” and the use of “starship,” whereas Justman introduced the idea of a “to do this, to do that” itemized list of what the Enterprise was up to in space. The phrase “where no man has gone before” is claimed by Sam Peeples, who wrote the series’ second pilot and used the phrase for its title.
The full story may be available in the Gene Roddenberry special collection at, I believe, UCLA. Going by the few pages devoted to the subject by Inside, I would guess that Roddenberry got off to a lousy start, collected phrases and ideas like a magpie, then pulled them together into the magnificent final product we now know. At least, if more credit could go to Black or Justman, Inside would definitely send it their way. Justman is the book’s co-author, and in regard to Roddenberry it has a pronounced “bad stuff ’bout” tendency.
A final point is that Roddenberry wrote some of Star Trek’s feebler episodes but is said to have done very good rewrites of other people’s scripts. The same bent can be seen here.