Series:
Steins;Gate: some of the best use of time travel. 24 eps
Cowboy Bebop: space western. 24 eps
The Tatami Galaxy: go in blind. dont worry about sub speed, slows down after first ep. 11 eps.
Shinsekai Yori: very complex sci fi. 24 eps
Code Geass: Gateway mecha anime. 50 eps. similar vein to FMA.
Haibane Renmei: Character driven slow burn and philosophical and Psychological. 13 eps
Serial Experiments Lain: mindfuck sci-fi. 13 eps
Texhnolyze: Dystopian sci fi. 24 eps
Bokurano Ours: Mecha with a twist. go in blind 24 eps.
Moribito: Like Avatar the Last Airbender but not as black and white. 26 eps.
Erased: gateway anime. go in blind. 12 eps.
Madoka Magicka: magical girls not for kids. Gateway anime for those who have the "childish stigma". 12 eps.
Eve no Jikan: great look at AI. 6 eps or a 2 hour movie.
Made in Abyss: Gateway anime, good for dispelling childish stigma despite cutesy first look. 13 eps.
Gurren Lagann: Over the top mecha wildness. May not dispel childishness, but its very inspirational. The flipside to Code Geass. 26 eps.
Twelve Kingdoms: If you like LOTR and GoT. Great characters and worldbuilding. 39+ eps.
91 Days: during alcohol bootlegging. great series. 11-12 eps.
Ghost in the Shell SAC: Great look at technological pervasiveness and ai. 50 eps
Anohana: if you want to be bludgeoned into feelings. 11 eps.
Devilman Crybaby: Definitely not for kids. 10 eps. go in blind! Its on Netflix
Violet Evergarden: First 4 eps may be slow but very emotional. Ep 10 is a great way to.see if you like it. 13 eps.
Movies:
A Silent Voice: About bullying.
Your Name: go in blind.
The Tale of Princess Kaguya: amazing animation and mature storytelling.
I want to Eat your Pancreas: Go in blind.
Milennium Actress: go in blind.
Akira: cyber punk.
Ghost in the Shell: cyber punk.
Patlabor 2: Great look at political geopolitics. You can watch the Patlabor series and first film, but this stands on its own.
Wolf Children: sounds Twilighty but go in blind.
Jin Roh Wolf Brigade: go in blind. will dispell "childish stigma"
Princess Mononoke: Least child friendly Ghibli film along side...
Grave of the Fireflies: bout ww2.
Perfect Blue: flip side to Milennium Actress. Inspiration for Black Swan.
Paprika: inspiration for Inception.
Maqia: if you want to cry.
Garden of Words: only 40 minutes.
Hope this helps.
Also seeing an entire medium if entertainment as one thing is probably nit the best way to look at entertainment. I think its great your willing to dip your feet in.
Hope you have fun.