Some YouTube links related to the 1960s ....
The first episode (also called the second pilot) of STAR TREK titled "Where No Man Has Gone Before" had an alternate beginning, with some different shots and music. Video link here. The music that was used in the title and end title sequences was released in a TOS soundtrack CD some years ago. There's an extra line of dialogue from Kirk to Spock in the teaser.
And here are links to some network previews of STAR TREK from 1966 - 1968.
The first promo may be the very first glimpse that anyone had of the show. It's about 20 seconds long, and consists of some different shots of a painting of the principals. A painting that was used on the the cover of the James Blish STAR TREK adaptation. Why "this first adult space adventure blasts off September 15th," I don't know. I mean, the first show was shown on September 8th, 1966. Did this broadcast of a week earlier not "count" since (I'm guessing here) it was part of an NBC premiere week? Maybe.
The second season promo is longer, and has a mechanical sort of music with some drumbeats. There is a slight variation of the promo here.
"I'm Captain Kirk of the Starship Enterprise. Join us each week for exciting, exotic adventures in outer space," says Shatner as he comes down the steps in the engine rooms and walks to the camera. This is the promotion for the third season of the show. There must have been no new footage (perhaps due to its last minute reprieve from cancellation), since the only exotic action that's shown is from the previous season.
And just for fun, here's one more video: a 1965 promo for the entire "all color" NBC line up. Like I said it's 1965, so there's no TREK yet -- but the first scene (which I won't spoil) is pretty prescient of what happens at concerts and plays and movie houses today. This promo runs about 25 minutes, and I didn't stay to watch it all.