The Star Trek Movie from 2009 was pretty good, but personally I
prefer Star Trek: The Original Series and the movies that were made
to go along with the original series.
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The best 'original' series episode to show someone who has no
idea about "Star Trek" is the episode entitled "The Trouble with
Tribbles". It captures the spirt of the original series
perfectly.
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The original Star Trek ended in 1969 as NBC cancelled the series
after just 3 seasons.