I'm going to break the mold a bit, here. There's a whole lot of stuff you'll want to watch, but there's also quite a few places where you could fall off.
For a modern viewer, consider what you're looking for. If you want fantastic adventure, begin with Season 2 of Star Trek: The Next Generation. First watch the pilot (Encounter at Farpoint) - it establishes the characters and will tell you 90% of what you need to know about them. Season 2 is where the show 'grew the beard' (and originated that term) and started getting good.
If, instead, you are more interested in character-driven drama, begin with Star Trek: Deep Space Nine. This show splits the Trek fans somewhat - they either love it or hate it. It centers around a space station, not a starship, and is less episodic in nature than other shows in the Star Trek franchise.
Once you've started down the road, you will likely end up watching everything Star Trek if it catches your interest. Rather than give you a viewing order, which others have already done, I'd like to give you a brief synopsis of each show and movie grouping.
Star Trek: The Original Series (TOS): The show that started it all. A five year mission, exploring deep space. The Klingons here look different from the rest of the Klingons you'll see in the newer stuff (especially the movies). It's also very obviously from the 70s - the sets, costumes, music, etc are all dated to the modern eye. Well worth watching, but go in understanding what it is: proto-Trek - the ideas are there, but it is limited by the technology and audiences of the time.
TOS-era Movies: General rule for movies 1-6: the odd numbers are nowhere near as good as the even. Star Trek: The Motion Picture suffers from being written like an episode, not a movie. II, III, and IV are a longer story arc involving easily the most tumultuous time in TOS. II is likely the best Trek film. V is likely the 2nd worst, but VI ends it on a high note and shows just how much some characters have grown and changed.
Star Trek: The Next Generation: This is what most viewers who've been with the show for a while think of when you say "Star Trek". The plots are more complex than TOS, the characters more fully realized, and the ship feels like the future. Ironically, many of the modern conveniences we have bear a striking resemblance to the things they have on the ship - datapads, communicators, even the ship control systems all influenced the generation of engineers that make our toys now. The first season is painfully stop 'n go as actors and scripwriters figure out what works and what doesn't, but once it gets going it doesn't slow down much.
Star Trek: Deep Space Nine: This show pushes us out of our comfort zone a bit. Especially in the early seasons, we're dealing much more with a few alien cultures in more than single episodes. The primary focus is more on characters and how they interact (until the later seasons) and it uses TNG's later seasons as a jumping-off point.
TNG-era Movies: Starting with Generations, moving on through First Contact, Insurrection, and Nemesis, the TNG era movies are a mixed bag. Insurrection and Nemesis are somewhat painful to watch (especially Nemesis, which is - in my opinion - extremely poorly written). Generations is the hand-off movie, with the core players from TOS featured, but focusing on TNG's cast. First Contact is likely the best movie from this set.
Star Trek: Voyager: You can enjoy this show. There's a very, very large and vocal portion of the fanbase which hates it. The first season suffers from a significant number of problems, many characters are never well fleshed out, and the only characters I really liked were The Doctor and 7 of 9. After the mixed reviews of DS9, Voyager put the show back on a ship. It isn't the Enterprise, and they wanted to break the ship away from the Federation. Lost In Space-style shows were popular at the time, so Voyager hopped on the bandwagon. Meeting new aliens and old enemies, Voyager must somehow defeat the odds and return home. The premise was good, but the show was easily the worst Trek show. It does improve towards the end, but not enough to save the show in general.
Enterprise: Set before TOS, before the Federation, this show focuses on humanity first exploring the stars. Then time travel gets involved, and established canon goes out the window. The show suffered from inconsistencies throughout it's run. Characterization, scriptwriting, character dynamics, nothing ever stabilized. It's a fun show to watch and it features a deeper exploration of the Federation's core races than most other shows have seen, but it failed to live up to the quality of TNG and DS9. It does, however, handily surpass Voyager.
Star Trek and Star Trek: Into Darkness: Abrams takes the reins. These movies are set before TOS, when the crew of the original TOS Enterprise are fresh from the Academy, long before their 5-year mission. Within the first 15 minutes of the first film, existing canon is dragged behind the chemical shed, shot, and thrown into a shallow grave. These movies, especially the first one, are pure popcorn flicks. They have the window dressing of Trek, they have the elements of Trek, but the first movie was missing something important. Into Darkness, however, realized it. They cared for the script more, they built a more believable story around what the first movie left behind, and it played well as a tribute to the best movie in the franchise (Star Trek II: The Wrath of Khan). The references were done well without seeming cheesy or forced, and they toned down the worst elements of the first reboot movie.
