I've never been what some people would call a "Trekkie" or "Trekker" or whatever you call a super-duper Star Trek fan. I've always leaned toward Star Wars (as many kids in the 1980s did).
I'm too young to have experienced the original series in the 1960s. My first introduction to the Star Trek Universe was when my dad took me to see Star Trek II: The Wrath of Khan. I loved it! I think I've seen nearly every other Trek in the theater since Khan and have enjoyed them all in different ways.
I vividly remember Star Trek: The Next Generation airing. Many of my friends and I thought it would be stupid (how could anyone replace Kirk and Spock's crew?), but we slowly started to enjoy them. Other than a year or so, I never really watched it religiously even though I thoroughly enjoyed the show.
Thanks to the wonders of today's streaming technology, I've been going through the show again, and I'm currently at the beginning of season 3. I don't know if it's because I'm older or I just enjoy ALL kinds of stories these days, but I can't believe how much I like this show!
Sure, there are some weaker episodes in the first two seasons, and some of the effects or sets might seem cheesy to today's audiences, but I urge you to stick with it and look past these shortcomings. Focus on the writing and the relationships between these characters. By season 3, they are really starting to come together. I can see why so many people love this series.
Anyway, I thought I'd mention this if you're looking for a new series to start binging. As someone who is just now in Season 3, I can say it just keeps getting better.