Take it slow and enjoy it. The original series has a lot of charm and quality to it. Begin with both TOS and The Next Generation. TOS episodes don't have to be watched in order, although they really should. TNG on the other hand, should be watched in order. TNG and Deep Space Nine were a highly respected platform with very high production value that allowed the actors, writers, directors and guest directors and fans to explore different genres such as action, suspense, drama, comedy, detective mystery, spy thriller, homage and classic trek with the familiar Star Trek universe. DS9 premiered half-way through TNG, and canonically at the same period of time as TNG. Personally, I watched TOS and TNG since I was a young child, and although I was aware of DS9 when it premiered on TV, and I had had some glimpses of it, I never watched any of it. This really helped me imagine what life would have been like for the crew of a starship like the Enterprise from TOS, or the Enterprise-D from TNG, and how crew members aboard the Enterprise-D would have been aware of DS9, although they would not have been on DS9. This was important for me because, like a crew member on the Enterprise, what life aboard Deep Space Nine would be like was left mostly up to my imagination. It wasn't until I was a young man that I watched DS9, so there's a significant time gap there, that separated the two for me emotionally. I was then able to imagine what life would have been like, at the same period of time in canon, but for the crew members of Deep Space Nine. Technically, all episodes of TNG and especially(!) DS9 should be watched in order. With DS9, strictly in order. And since DS9 premiered half-way through TNG's run, it would be more interesting to be aware of how that played out for audiences at the time, as there are canon events that affect both shows' story arcs. I strongly recommend you enjoy TNG and DS9 the way I did, by finishing TNG and then starting DS9 later. The TOS films can be watched whenever you feel like it, they are excellent Star Trek. Technically, Star Trek III: The Search for Spock should be seen before Star Trek VI: The Undiscovered Country, but it's not really that important. Star Trek III takes place immediately after Star Trek II: The Wrath of Khan and they should be watched in that order, however, I strongly recommend against watching them back-to-back as they are not meant to be seen that way. They are each directed by two different people and represent different styles of Star Trek, with STII having the feel of an action movie and STIII having the feel of the original series, with a different pacing that long-time fans are more likely to appreciate. I have recently watched all the TOS movies, and I left out Star Trek V: The Final Frontier because I didn't want to taint the experience - it's reputation is very bad and you may want to avoid it until you are really curious. The TNG movies should be watched in order, after finishing TNG; and, since they premiered while DS9 was still running, could be watched relative to the episode run of DS9 to catch subtle references to canon events between the movies and the DS9 episodes. There's really no need to bother with Voyager or Enterprise at all, they're really not that good. The animated series can be enjoyed for fun whenever you feel like it, it was good! Just don't rush it, because you enjoy Star Trek much more when you become familiar with the canon and comfortable with the fiction, so that the fiction doesn't get in the way of the storytelling and emotional underpinnings of the canon episodes, however, you can also appreciate the art episodes that are more about what you can do when you take a familiar story and add a stylistic twist.
