If you're starting as a beginner, I do think it's important to see some of the original series (called "ST TOS" by the in-crowd) because that establishes the basic structure, characters, organization, etc., that is followed throughout all the different series and movies. I'd track down a few of the best episodes in that series and watch them as a great way to start. I'd recommend:
The Ultimate Computer
Balance of Terror
Space Seed (basis for the later Khan movie)
The Changeling
The Galileo Seven
City on the Edge of Forever (probably the most famous episode; rather more drama than science per se, but built around a time travel paradox theme)
Welcome to a wonderful universe. 
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