edit: feel free to reply no matter how old this thread gets!
I'm not particularly into anime, I like it just as much as live action shows or other types of animation. Having said that, I feel like watching some and I'm not sure what to pick.
Here's a list of other anime shows I've watched/manga and novels I've read, so you can better judge what to recommend. ( would greatly appreciate the story to be completed, I hate waiting years for closure) :
Full Metal Alchemist, Naruto : both great stories, I've read the mangas and loved them. They wouldn't be a priority to watch because I already read them.
Attack on Titan: started with the anime, it was great, switched to the manga, got extremely disappointed with the story development and its ending. It felt a bit like nonsense for me.
Dragon Ball/Z : probably my favourite, I've rewatched it recently. I don't like GT and Super.
Overlord: I've read good part of the novel and manga. I'm liking it so far
JoJo's Bizarre Adventure: please No thanks. Not for me.
Death Note: loved the first arc, not a fan of the ending but no hate for it unlike with AOT.
I also recall Escaflowne and Inuyasha, they were alright but not something I've loved or hated. Cowboy Bebop also in this category, excellent visually and great OST but the story wasn't as gripping.
Other than that... Feel free to suggest any other titles! Extra thanks if you are specific about any versions and why is the show so great.
I'm also not a huge anime fan, so my interests in the genre are pretty selective. The anime shows that I tend to enjoy are ones that usually aren't ongoing stories, and wrap up their narrative lines within a season or two. That said, I absolutely loved Hellsing and Hellsing: Ultimate (which is a sort of "reimagining" of the original series, if I recall correctly). They've got some of the most entertaining action scenes I've seen. The CastleVania series on Netflix is also pretty fun, though it starts to get pretty weird in the later seasons (though I'm feeling hopeful for the new CastleVania series that they just teased). I've also recently enjoyed Chainsaw Man, and am pretty excited for the next season to come out.
I also highly recommend Redline. It's not a series, but a standalone film, and it's one of my favorite movies ever. If you're at all familiar with the F-ZERO games, think of that, but as an anime.