Corbomite Maneuver hands down. Journey to Babel is another classic but without knowing the back stories it is not as strong. Balance of Terror is great but it is really an old fashioned sub vs. destroyer flic.
Corbomite has it all. It introduces you to the crew, has tons of action, and gives you a lump in your throat morality lesson at the end.
It is the best of Trek and a great intro.
Break open a bottle of Tranya and enjoy.