Star Trek is more than a SyFy Show

When I was very little - I'm talking two or three here, - my family had very little money.

My dad was posted overseas and would often be in the field for days at a time, leaving me and my mom basically alone in a foreign country. 
Sometimes Mom would read to me from anything she could get her hands on - books far above a child's reading level, let alone understanding, - but other times we'd watch taped episodes of TNG her parents had sent from the States. The same episodes, over and over again. 
I think I imprinted on those classic, Gene Roddenberry Star Trek ideas: Humanisim. Optimisim. Muliculturalism. Multi-species-sim. The power of science. The Federation. The belief that, somehow, humanity might manage not to destroy itself.  
Such was it's influence on my early life that, when I started school and time came to learn "The Pledge of Alliegence," when the teacher asked if any of us already knew it, I stood up and recited Kirk/Picard's "These are the Voyages..." speech instead. 
Don't get me wrong. I love dark!science fiction, perhaps more so at times than the bright optisim of Roddenberry's STs, but none of them have influenced me, as a person and writer, as much as TNG. And, sometimes, I just wish that other people would remember these things - that humanity is not neccessarily evil, that we will not neccessarily destroy ourselves, and that apocalyptic futures are not the only ones that might come to pass.

I want to watch Star Trek: TOS, but I've seen an episode and it was so over the top dramatic that I spent half the time torn between cringing and laughing. So I'm not sure it'll go well.

I love the story about you and your mom watching TNG episodes together - I think that's really sweet.
TOS is over the top and dramatic and just awful at times, but the ideas are worth it.

Try "Where No Man Has Gone Before" or "Balance of Terror"
I've heard "Where No Man Has Gone Before" mentioned a lot. As well as "Amok Time" and others that I can't think of at the moment...
"Amok Time" is pure unbridled over-the-top dramatic nonsense, with a sprinkling of what can be considered the greatest case of Kirk/Spock UST in all of ST.