I was born in 1986, the year before The Next Generation premiered as the second installment of Star Trek on television. So I have some very early memories of episodes in the later seasons as it continued it's run. Then Voyager came on when I was in junior high and I was hooked from the beginning. My Trek obsession really began with Captain Kathryn Janeway and her crew's struggle to get home.
Some time later I discovered episodes from The Original Series on VHS at our local library. I burned through as many as I could in short order before getting the entire series on DVD a few years later.
And then there was Deep Space Nine, the redheaded, forgotten stepchild of the Star Trek Universe. I never got into DS9 because my parents never watched it. I'd always loved the Trek novels, and inevitably that involved some advances by DS9 characters as I started to run out of Voyager and TNG material. With so many of them intriguing me (mainly Ezri Dax and Julian Bashir), I couldn't help but pick up a few seasons of DS9. That was a good call.
Deep Space Nine is without a doubt the best Star Trek series, and it isn't even close. Voyager wins the prize for best episodes, but DS9 is the best series as a whole. And the reason for that is twofold. One, it has the best cast. Two, it does the most to create and expand the Star Trek Universe.
Granted TOS had an amazing cast...but there's only so much Kirk boinks an alien, Spock makes a pithy joke, and McCoy gets angry at Spock I can handle. The Next Generation had probably three or four of the worst characters in Star Trek: Wesley Crusher, Beverly Crusher, Deanna Troi, and Katherine Pulaski. Voyager had a few characters who were boring (Chakotay, Kes), and a few characters that had extremely obnoxious story arcs (Neelix, The Doctor).
In Deep Space Nine, there wasn't a single character I didn't love. Avery Brooks had such a jovial and laid back approach to captain Sisko, he was a joy to watch. I will never tire of seeing Nana Visitor kick someone's ass. Terry Farrell had an amazing, mercurial, and ultimately wise approach to Jadzia Dax, and Nicole de Boer deserves a medal for taking that blueprint and creating something else entirely. Siddeg Al Faddil was amusing in his smarminess, and Colm Meaney was a perfect companion for him. The only places the series suffered were where Michael Dorn couldn't do anything to liven up Worf, and the few too many trips to Ferenginar. (And not murdering Keiko O'Brien immediately.)
What draw people to something in fanatic levels is a universe they feel they can immerse themselves in. It's why The Lord of the Rings, and Harry Potter are so popular. DS9 did more for the trek universe than any of the other series combined. We learned about the Romulans, the Klingons, the Cardassians, the Bajorans, the Tzenkethi, the Breen, the Dominion, the Ferengei, the Nausicans, and even got the best look into the Federation itself of all the series.
TOS and TNG are important because they (in different ways) mark the beginning of the Star Trek Universe. Deep Space Nine, in returning to the roots of Star Trek, helps it's legacy to endure. Where The Original Series helped shatter the race barrier and diversify television, Deep Space Nine started to deconstruct some of the taboos we have about gender and sex. Where TOS featured the first inter-racial kiss on television, DS9 featured the first same-sex kiss on television. In DS9 we broke from the safe world of TNG and entered an area where no topic was too dark. Sex, Death, Genocide, Religion, nothing was sacred.
Deep Space Nine looms as the greatest Trek Series because it grew to be larger than life in a way that TOS was too far ahead of it's time to accomplish.