Yeahhhhhhhhhh stuff I could recommend that's recent:
Puella Magi Madoka Magica - Deconstruction of the magical girl genre, deals with dehumanisation and suicide and such. Exploded everywhere now. 3 feature length films as well as the anime series now. Two spin off mangas too.
Psycho-Pass - Dark dystopian mindf*ck. Computers read people's brains and determine your emotional state represented via a hue/colour and possibility of you committing a crime by a number. These affect everything in your life. Once you reach a certain state you're deemed emotionally broken and police will hunt you down. Enforcers (convicted criminals aiding the police) work with the police to help since being emotionally damaged by the job isn't relevant once you're "broken". Sounds like minority report, really isn't. One of the best of recent times.
Attack On Titan (orig. Shingeki No Kyojin) - VERY gripping. Humanity has been close to wiped out/eaten by "Titans". Huge humanoids with no sex organs or digestive systems. Ignore all other life, just seem to chew up humans for the fun of it. Starts with 100 years of the last human colony being all fine and happy.
Future Diary - God of Time and Space modifies a few people's diaries to tell the future (details of the future reflecting the person's diary keeping style, future can be changed easily). These people have to battle royale and there has to be only one winner/survivor or the universe dies, winner gets to be the new God too.
Ummm so those are all great. I checked out a lot of more mainstream stuff once upon a time. One Piece has an amazing convoluted plot with lots of details but I like that kinda stuff so it's fine and dandy with me. Toriko's the other one I still keep with. Not much else though in that respect. Dragon Ball was amazing as a child/still is.
I suppose if you want to dip your toe into stuff/explore you have to bear in mind there's a few different areas. The films are a lot more versatile than series' a lot of time since a manga or anime series needs to be kept going (though concise short anime series that end).
AIMMAR'S SHORT GUIDE TO MANGA/ANIME SERIES DEMOGRAPHICS
ANYWAY so there's Shonen (roughly means "young") stuff which is often in the form of "battle" manga. This has a young male demographic and notable examples are: Dragon Ball/Z, One Piece, Bleach, Naruto. (I don't get Naruto at all though).
There's also stuff denoted as Shojo (young female demographic) which is mostly romance manga/anime which is also very popular, a lot of them being set in schools.
Then there's Seinen which is marketed towards older males mainly, can be anything from warped mind-bending experimentation to just near-pornographic nonsense. Think there's a female equivalent called Josei (?)
Not everything you encounter is locked into these boxes obviously but they're helpful guides. Most popular genres are battle, sports, romance and comedy but you get virtually anything.
now...
AIMMAR'S SHORT GUIDE TO ANIME FILMS
Pretty diverse area, so you can get anything really. Well known recommendations right off the bat are:
Akira - Futuristic dystopian cyberpunk beauty. Masterful animation. Incredible for the 80s.
Steamboy - Large scale steampunk themes, set in victorian England, very well crafted.
Paprika - Dream based frolicking, probably inspired Inception, captures how we exist in dreams well too.
The Girl Who Leapt Through Time - Made by the Summer Wars guy. Schoolgirl discovers how to time travel and just does relatively mundane things with it but lovely animation. Quaint? I'm not that good with language/describing.
Ghost In The Shell and Evangelion have been mentioned and they're radical too. They have manga, various TV series and films though. You need to mine those franchises because there's a bunch of good there.
ELEPHANT IN THE ROOM: Studio Ghibli/Hayao Miyazaki
This guy (Hayao writes/directs most of these) and his studio make some of the best known anime films. Well animated and immersive worlds that are well realised and draw on all sorts of things but with a nice spattering of older japanese folklore/religion etc. Endearing characters, truly cathartic and you'll get a whole boatload of feelings from any of them.
You might have heard of: Spirited Away, My Neighbour Totoro, Princess Mononoke, Howl's Moving Castle
Other notable mentions: Kiki's Delivery Service, Porco Rosso, Nausicaa of the Valley of the Wind
Really just watch all of them though, if you're starting out, the first two I mentioned are a good place to go from. You might find Ponyo a bit too happy though and a few of the ones I didn't mention were not by Hayao and noticeably have some magic missing.
AAAAAAAAAAAAND yeah I guess I'm done, off to top of my head some more good manga/anime I could recommend are:
Trigun - Space western kinda deal. Humanity's on a desert planet, sharpshooter with a murky past/alleged mass murderer wanders round trying to help people as he goes.
Hajime no Ippo - I never thought I'd be one for any sports themed manga but this is really well written and engaging. Boxing anime/manga. Special.
FLCL - Boy with a portal in his forehead gets trained to save the world from a looming threat. Interesting pretty short series. Nice sci-fi elements. Comedy angles but rather serious plot.
Cowboy Bebop - Sci-fi bounty hunters travel about space trying to resolve issues from their past as they go along. Relatively short but really well done. 26 episodes. Also excellent jazz soundtrack pops in now and then.
Outlaw Star - Eeriely similar to Cowboy Bebop, 26 episodes of a lovely space opera. Jack of all trades and his young sidekick become bodyguards for a girl then steal a spaceship with her and have crazy adventures. Gets all reality bendy towards the end. Great.
You may have noticed a lot mentions to futuristic stuff, welp in 80s-90s japanese anime was full of it and it's wonderful to look at.
Kay I'm done for now. Going to go back and make all my recommendations bold.
Or not.. yeah other obvious ones from my childhood I just remembered that all had token merchandise to buy into were: Pokémon, Digimon, Beyblade, Yu-Gi-Oh.
The last one is an excellent card game.