Soz
You're f*cking grounded.
If you haven't watched any Studio Ghibli movies yet, I definitely recommend those. There's a whole bunch of them! My personal favorite is Castle in the Sky which might be a pretty good gateway because it hews pretty close to a traditional adventure story. Honestly pretty much the entire studio's output is at the very least worth watching. Most of the movies are total classics and there's only a couple duds among them.
Since you're a total newbie to the medium I could point you in so many directions. If the concept of magical girls is at all appealing to you, check out Sailor Moon and Card Captor Sakura. They're both pretty lengthy classics. Sailor Moon has a recent 'remake' but from what I understand it isn't as good as the original anime. I haven't seen it though so I couldn't give my opinion. It might be a decent gateway though if you want something that looks more modern.
I also recommend Hunter x Hunter. It's a solid adventure show with some really unique characters and the "power system" is really fascinating. There's a lot of variety in the story arcs, too. The second to last and longest arc has some weird pacing quirks but they didn't bother me too much.
ERASED is a short but sweet show that centers around a man with the ability to step back in time right before deadly events occurs. At first he uses it to save people from being hit by cars and stuff, but an opportunity quickly unfolds to undo a terrible tragedy from his childhood.
The Devil is a Part Timer! is another short one, this time a fantasy comedy centering around a demon from a fantasy world getting stuck in modern day Japan. He gets a job at McDonald's (well it's called MgRonald's but it's pretty much McDonald's in everything but name). Hijinks ensue, and it turns out the troubles of the fantasy world follow him into Japan too.
Steins;Gate is another time travel show, this time about a bunch of nerds that accidentally invent a way to send text messages back in time. It takes a while for the story to get rolling but when it does it hits hard.
The Vision of Escaflowne is a very, very 90s classic about a girl who gets transported into a fantasy land. It has amazing music, very distinctive character designs, and a plot that gets totally bonkers towards the end. It feels like the essence of a 90s JRPG distilled into an anime show. Also, there's fantasy mechs.
If you want more movies, anything Mamoru Hosoda or Satoshi Kon (RIP, taken from us too soon) directed is worth a look for sure. Makoto Shinkai is a very popular director too but I actually haven't watched any of his stuff other than the short movie Voices from a Distant Star.
I agree with everything here except for their opinion on Steins;Gate. I was hooked within ten minutes. I personally see it as the best writing in any anime ever. Period. If you know anything of real world conspiracy theories then you'd get more out of it, but it by no means a requirement as they explain pretty much everything.
As for recommendations Assassination Classroom. It's a comedy-drama, and it's very easy to digest. I'm not sure if you consider this a boon or bane, but it isn't extremely short at two seasons of twenty something episodes each. This is an anime I recommend watching in English as the transliteration is better than perfect. They did a great job, and Sonny Strait is among my favorites for English VA. There are a ton of other well known actors here, too as the cast is enormous, but don't be overwhelmed by my review. It's a really simple story.
Another movie that just recently came out that is AMAZING is Kimi no Nawa (Your Name is the official English title). It's currently the 4th highest grossing movie in Japan ever only with Spirited Away, Titanic, and Frozen beating it which didn't beat it by much in terms of revenue. I -highly- recommend it, and will forever recommend Makoto Shinkai as a name to find good movies for. Previous work he's done got him know for excellent visuals and this movie has it's fair share.
If you're into comedy, but want something shorter than Assassination Classroom, I recommend Outbreak Company. Another one I recommend in English, as Tyler Galindo is an excellent lead character. More from Tyler would include Btooom! which is a much, much, much darker anime that I recommend equally as much in English if you're looking for something that will keep you on your toes.