Oh, I've just gotten started on these, too! As a second generation 'Trekkie' with my mother braving the fandom before me, I grew up watching the newer shows as a child, when they really were brand new. I can still remember the times when I thought Deanna Troi was the most beautiful woman on earth... or in space, I suppose, to be accurate.
Now, I'm a bit more grown up than I was then and somehow got roped into watching the first few episodes of TOS with my family on the xBox with our Netflix. I'm a little over half of the way through the first season, though I broke order to watch 'A Piece of the Action' because a friend told me it was funny, which it was. I have been very impressed by the hilarity of TOS. "Spock is so spunky!" is usually all I have to say when my family asks me how it is going.
I'm very glad to know I'm not the only person watching it through for the first time. I thought for sure I was the last one on the planet!