Animation is a beautiful tool that allows creators to tell wonderful stories. Just because these stories are from a culture different than what you're probably used to doesn't mean you shouldn't give them a chance!
FLCL
https://www.kotaku.com.au/2016/03/adult-swim-is-making-a-sequel-to-flcl-the-coolest-anime-ever/
If you're a fan of colorful and beautiful animation that tells multiple stories, this one is for you. The first episode aired in 2000. Lucky for us, Adult Swim has recently renewed FLCL for a second and third season. This is an anime that can be watched and rewatched in order to gain new appreciation for characters and little details.
Steins; Gate
http://steins-gate.wikia.com/wiki/File:Steins_Gate_-_06_-_Large_16.jpg
Steins; Gate was beautifully adapted from a visual novel. Every single character has a unique personality and something to add to the story and progression of the plot. When watching for the first time, I laughed, I teared up a bit, and I was glued to my screen in suspense and excitement wondering what was going to happen next. Steins; Gate is a sci-fi action with comedy and hints of drama. Although it is a bit slow to start off, getting to episode 12 is like a reward as things immediately pick up speed and things really get interesting.
Death Note
http://freakingeek.com/death-note-lanime-culte-coffret-blu-ray-collector/
You've heard of this one before. There's a live-action adaptation on Netflix. Don't watch that. Watch the original, it's better, I promise.
Ping Pong The Animation
https://www.imdb.com/title/tt3592032/mediaviewer/rm670119936
If you can learn to love the unique style of animation, you can learn to love Ping Pong The Animation. This is by far the most relatable anime imaginable. Were you a high school athlete? Have you ever done anything you loved to do competitively? If you answered yes to either of those questions, I promise you, you will relate to and thoroughly enjoy this anime. It's sad and it's happy. It has the capability to make you laugh and bring a tear to your eye.
Attack On Titan
https://www.geek.com/anime/attack-on-titan-designer-shares-season-3-cast-sketch-1718813/
You've probably heard of this one, too. Attack on Titan is fun to watch and keeps you on your toes when you think you're about to guess what happens next.
Sword Art Online
https://www.netflix.com/title/70302573
If you die in the game, you die in real life.
Interesting ideas and colorful animation
Cowboy Bebop
https://www.shoanime.com/resenas/cowboy-bebop-una-obra-de-arte-hecha-anime/
A true classic. Cowboy Bebop is an anime you don't have to watch in its entirety to enjoy. You can watch an episode here and an episode there out of order and still enjoy the plot progression and characters.