For someone that's only seen the new movies, where should I start? DS9?
I would start with
The Next Generation, from which
Deep Space Nine and
Voyager sprang. TNG gives you the background of and sets the stage for about 500 episodes between the three series, produced over a period of about 15 years (1987-2001)
TNG is set about 80 years after the original series. All you really need to know about the original series to bridge the gap between it and TNG you've probably already got the gist of from seeing the reboot movies.
The first couple seasons of TNG are admittedly not great. There are some good episodes, for example "The Measure of a Man" and "Elementary, Dear Data", and some sow the seeds of future story lines (particularly "Q Who" and "The Emissary"). On the whole they're pretty 'meh' and the second season finale, "Shades of Gray", is a colossal piece of shit.
The third season of the show upped the ante quite a bit. "Who Watches the Watchers", "The Defector", "Deja Q", "Yesterday's Enterprise", "The Offspring": there are easily more good shows in the third season than the first two combined. The third season is capped off with "The Best of Both Worlds", one of the better known and most revered episodes of the series.
From there on it's pretty good. DS9 began about halfway through TNG's sixth season (TNG had seven seasons total, as did DS9 and Voyager). To appreciate DS9's pilot really requires having seen "The Best of Both Worlds" Parts 1 and 2 (Season 3-4), "The Wounded" (Season 4), "Ensign Ro" (Season 5) and "Chain of Command" Parts 1 and 2 (Season 6) from TNG.
DS9 followed a similar track as TNG did: mediocre start before really getting good. As with TNG there were a few gems in the first couple seasons, particularly "Duet". "Necessary Evil", "Crossover", "The Wire" are also pretty good and set up later stories. The second season was the first one that had an ongoing story arc in it, with the first three episodes of the season leading one into the next. The second season of DS9 ends with "The Jem'Hadar", from which the rest of the show's story follows and which if you really pay attention to the foreshadowing throughout the season will realize is the culmination of a season-long arc. "Shit gets real" after that episode and in later seasons.
DS9 is the one Star Trek TV series where the characters are not all happy, shiny, perfect future people and has a lot more to say. It's my favourite and I think it was the best Star Trek series.