You've already got more advice than you can handle, but I'd recommend starting by sampling episodes of TNG and TOS.
Neither show is very arc-based. There are some ongoing storylines in TNG, but you can tune into any episode from any season and be just fine. You don't need to know any character backstories to get it or anything. TNG is basically the template of all the shows that followed, so it'll give you a good sense of what Trek is like.
Then you can check out some TOS episodes. Obviously, since it's the first, TNG takes its cues from here. But since it's much older, it's also more dated, so if you see TNG first it might be easier to swallow. And you can see how the story telling evolved between the two. Also, every episode is completely standalone, much moreso than TNG, so you can sample from any point in the series and be just fine.
Get a sense of what those two shows are like. These are basically the purest forms of Trek; they're both a ship called Enterprise exploring the universe. The subsequent series had to find different twists to distinguish themselves.
When you watch DS9, try to watch it in order, since for the most part, it tries to tell an ongoing story.
With Voyager, watch the pilot to get a sense of what the show was supposed to be about. After that, don't worry about continuity; it's inconsistent. There's less of an overall story, so you can basically pick up any episode, like TOS and TNG.
ENT is kind of similar to VOY--at least for the first two seasons, you can just watch any episode. The third season is intended to be a full arc though, and I'm not as familiar with the fourth season, except that it's the one that tries really hard to do its job as a prequel, so it's probably better to see it after you're familiar with all the other shows.
Also, make sure you watch all the movies, especially the TOS ones.