I saw the original series first back in the 60's when it was first shown on UK tv. I've since seen it several times and although it seems very clunky and dated, some of the stories still hold up.
I'd mention The Menagerie and The City On The Edge Of Forever as two of the most memorable.
The follow on series never really grabbed me but I did quite get into Voyager; the basic plot line of returning from the other side of the galaxy was interesting.
I've still not seen all of it, like many US tv series there were too many episodes made with an eye on syndication.
Loved both the 'reboot' films though.