Now is a GREAT time to dive into Star Trek! There's so much of it and it's available on multiple platforms, but you do need to understand that some of it is crap. It's a burden all Trek fans carry. 
My personal guide and recommendations for viewing all things Trek is below:
TOS (The Original Series): For a modern viewer, some of it is tough to watch, and not because of the special effects. For something made in the 60's, it's phenomenal, and quite a few episodes hold up well, but there's barely any continuity other than they are on a Starfleet vessel. It's basically a "Space Western". What you watch it for is the dynamic between Kirk, Spock, and McCoy. The three of them are AMAZING. Scotty also interacts with them brilliantly. You can definitely hand-pick which episodes to watch and skip some of the stinkers. Favorite episodes: The Doomsday Machine, The Trouble With Tribbles, Balance of Terror, Space Seed, and Mirror, Mirror.
The Animated Series: You can honestly skip this one. It's fun to watch if you're nostalgic for it, but otherwise, move on....
Now would be a good time to watch the original 6 movies. They are all pretty decent in their own ways.
ST: The Motion Picture: This one was a real big budget movie with special effects that hold up well even today. A lot of that budget was spent getting "beauty shots" of the refitted Enterprise. The story involves the crew of the Enterprise (with some new additions) facing down an unknown space probe that is destroying everything in its path. It can be slow at times, but still worth watching.
The Wrath of Khan: Oh man, this one RULES. You've seen the memes. It is a re-visitation of an old Original Series episode Space Seed with the evil Khan, played by the ever-classy Ricardo Montalban. No Rich Corinthian Leather here; he's a ruthless bastard of a villain. Even if you hate Trek, this is an excellent movie.
The Search for Spock: This one is just ok. They search for Spock! A Klingon commander named Kruge (played by Christopher Lloyd) complicates things. They steal his ship and make it a time machine. Yes, seriously.
The Voyage Home: This is a Save The Whales! bumper sticker in movie form. Kirk and the gang have to time travel back to 1987 San Francisco in a Klingon ship stolen from Christopher Lloyd to snag a pair of humpback whales named George and Gracie and bring them back because there aren't any whales left in the 23rd Century and a giant space turd/probe showed up to Earth looking for whales and is literally ripping up the oceans to find them because it's PISSED. Yeah. This one is a ton of fun. Some really funny moments here.
The Final Frontier: Shatner directed this one and it's mediocre, but watch it anyway for the Kirk/Spock/McCoy dynamic. A Vulcan hippie shows up and steals the Enterprise to find God, who lives in the center of the galaxy. Weirdness ensues.
The Undiscovered Country: The last original series movie, and it might just be the best. Conspiracy, intrigue, and Klingons abound. And a Shakespeare-quoting Klingon general played by Christopher Plummer. Oh, and Kim Cattral is a Vulcan officer in the middle of everything. A perfect send-off to the original crew.
Ok, back to the TV stuff.
ST: TNG( The Next Generation): There's 7 seasons to watch, and you might want to start here. It takes place roughly 75 years after the last Original Series movie on a much different Enterprise. A common theme with most modern Trek shows is having a garbage 1st season, and TNG starts this trend. Season 1 is ROUGH, but still worth checking out. Season 2 ditches Dr. Crusher for a year for universally hated Dr. Pulaski, but Crusher returns for Season 3 and stays until the end. I just finished a run through every episode of this series, and it is one of the best, if not THE best. The crew is great, Picard is a great captain, and while they do have many canned episodes, they start to get more serialized toward the end. There are some fantastic stories here, starting in Season 2. Character building is excellent, and things that happen on this show lay the groundwork for everything else. Favorite episodes: The Measure of a Man, The Royale, Booby Trap, Q Who, The Best of Both Worlds, The Inner Light, Chain of Command, Starship Mine, Descent, Parallels, The Pegasus, and All Good Things... To be honest, most of it is great. Watch it.
Deep Space Nine: Now, we're getting good. This is probably my favorite series. Heavy serialization starts here, and the characters are very interesting. You want epic space battles? You'll get them here, especially later on. Again, 7 seasons of material here, and the first two basically set up the rest of the show; season 3-onward is some of the best TV ever made. Worf shows up from TNG, and when he does, business really picks up. Not everything is happy, shiny, cruise ship Trek like TNG, either. Most of it takes place on an old alien space station inherited by the Federation, and it's a real POS until they fix it up. Eventually, they get a sweet little battleship, which expands the show. It's without a doubt the darkest series, and some parts will rip your hear out. Favorite episodes: The Jem'Hadar, Past Tense, The Sword of Kahless, Trials and Tribble-ations, Who Mourns for Morn?, In The Pale Moonlight, Tears of the Prophets, The Seige of AR-588, What You Leave Behind
Voyager: This time, its back to a ship, but this ship is strewn across the galaxy and they have to figure out how to get home. Another 7 season show, and people like to trash it, but it has some great content. First season is slow, but it picks up a few seasons in. It has sort of a Battlestar Galactica (2004) feel to it, albeit a less depressing one. Some familiar enemies eventually show up, and it finishes strong. Favorites include The 37's, Future's End, Year of Hell, Equinox, and Endgame.
Movie time again! The Next Generation crew got four movies as well. Watch them here:
Generations: This one hands the baton from the original crew to the TNG crew. It's decent-ish, but watch it anyway.
First Contact: This one is the best of the TNG crew movies. The Borg show up to try and alter Earth's past and make it so the Federation never existed. Picard & Friends go back in time to stop them, and end up having to help the guy who created Warp Engines on Earth launch the first warp ship so the Vulcans see them and make... ahem... FIRST CONTACT. Assimilate THIS! By far the most action packed TNG movie.
Insurrection: I'm gonna be honest, this one is a stinker. It could have easily been a TNG episode. Some alien guy wants to live forever and some planet knows how to, so the crew of the Enterprise gets involved because the guy is a jerkface. And Riker steers the ship with a PC joystick. That's all I got.
Nemesis: A guy (played by a young Tom Hardy) takes over Romulus, which is BAD. He reaches out to Picard, and tries to destroy him and upset the balance of power in their part of the galaxy, so naturally he needs to be stopped. Along the way, the crew finds a Data prototype named B-4 (because he was made B-4 Data, get it?). Also, Picard has a sweet space buggy with guns. It's panned, but I liked this one, mostly.
And back to TV...
Enterprise: This one was a little different than the rest. It takes place 100+ years before the original series with Kirk, Spock, and McCoy, before the Federation even existed. Humans build a "Warp 5" ship and go out into the galaxy exploring in earnest for the 1st time. They don't know a lot of the aliens we have come to know yet, and their best allies are the Vulcans, because Humans met them 1st. Their relationship is rocky at best because they think of Humans as primitive. They send one of their officers on the ship to "observe" but she becomes a trusted and valued 1st Officer to Captain Archer, played by Scott Bakula. Think of it as a 4 season long Quantum Leap episode and it makes sense. It's good, but not quite as good as the earlier shows. I'd watch them all; it's relatively short because it got canceled prematurely. It was getting good at the end, so this is sad.
Discovery: This one takes place 10 years before The Original Series to start. It's controversial because everything looks like it's in our future, not the cardboard sets like something made in the 1960's, so it rustles A LOT of jimmies. But, it's still Star Trek, and it's worth watching. There are two seasons so far, but Season 2 is really great.
Optional Viewing: The JJ Abrams Movies. I enjoyed them, but they take place in a divergent, alternate timeline due to Spock and a Romulan mining ship called the Narada going back in time and changing history when the star in the Romulan System explodes, which destroys Romulus. Spock is trying to stop it from going supernova, but fails, and both ships are dragged into a singularity that sends them back over 100 years. Nero, the captain of the Narada, is PISSED and wants to kill anything and everything Federation because he blames them for the destruction of Romulus. He starts by killing Kirk's dad.
JJ Trek Star Trek: This one starts with Nero killing Kirk's dad. This changes everything. The direction of Starfleet changes, Kirk turns into a punk kid with no desire to join Starfleet, and has to be recruited by the captain of the Enterprise: Christopher Pike. Except in this reality, the ships are bigger, more powerful, and look a little different than what we are used to. The OG crew, a lot younger here, somehow ends up on the ship. The pieces are there, but everything is different. Then, Nero comes out of hiding and they are called to action, galvanizing the crew. It's pretty good, and has callbacks to the stuff we are used to.
JJ Trek 2: Into Darkness: Eh. Khan is back. Except he's played by Benedict Cumberbatch. Did I mention Khan is supposed to be Indian? Odd... Anyway, Peter Weller's also here, and he's a militaristic weirdo who created a gargantuan ship called the Vengeance that's actually pretty cool. The rest is forced action and hijinks that is just meh.
JJ Trek 3: Beyond: The crew has been out in space for a while on their "5 year mission" just like they did on The Original Series. Kirk is bored. Eventually, an enemy shows up in the form of Idris Elba and really screws things up. This one is the 1st of the JJ movies that galvanizes that relationship between Kirk, Spock, and McCoy, and it feels GREAT. I liked this one a lot (except for one scene involving Sabotage by the Beastie Boys), but it kinda bombed at the box office, so we're still waiting on another JJ move. Also, JJ hopped onto Star Wars. Maybe we'll get another one now that he's done with that.
And that brings us to Picard. The things that happen that diverged the timeline cause irreparable harm to the Prime universe (which is the non-JJ stuff). Picard picks up 25 years after that went down. Only one episode in, but this is gonna be good.
TL/DR: Watch all of it. It's great.
