Yeah you can stream the series on Netflix (TOS-R), Amazon Instant Video (both), and Hulu Plus (TOS-R). If you've never seen it and you want to give it the best possible chance, watch the new effects versions. They will be easier to take for you and give you less to find fault with. I mean, if you already don't like the music, you won't like the 60's effects. I would only start new, younger fans on the originals effects if they are VERY young and won't care, or show a preference or liking for any era's technology (for example, people who enjoy the 1933 King Kong and don't look at it as dated at all). Since they don't alter the story, and in some case help them (Tomorrow is Yesterday), just go for the new effects.
As for Enterprise, it was a good series. The first season was weakest, but it improved as it went on. The third season was great and the fourth season was a lot of fun as it tried to fill in the gaps between that series and the original. It deserved three more years.
Voyager, I didn't care for. DS( was mostly excellent. TNG was great at the time, but it is the lest "rewatchable" for me. Too talky and stuffy a lot of the time with technobabble A & B plots.
Good luck, I hope you like it. If not, I give you credit for trying it.