Seirei no Moribito. It's about a female mercenary who is hired by the empress of medieval fantasy Japan to take the prince and go into hiding to protect him from being assassinated by his father the emperor, who thinks the prince has been possessed by a demon that will destroy the realm. Adult protagonist. Has a good soundtrack by Kenji Kawai. 24 episodes.
Fate/Zero. In an urban modern day city in Japan, seven mages summon seven heroes from ages past to fight to the death in a secret tournament for the Holy Grail, a magical artifact that is said to be able to grant any wish. Very atmospheric, heavy casualties. Adult protagonist. Plot is pretty good, it was written by Gen Urobuchi. Memorable soundtrack by Yuki Kajiura. English dub is fine. 24 episodes. Fate/Zero (along with SAO and AoT) is what made anime mainstream, and popularized the Fate franchise.
Legend of the Galactic Heroes: follows two protagonists on opposites of a galactic war: a noble rising through the ranks of space Prussia and becoming a benevolent dictator, vs a historian who is forlornly fighting for a corrupt and decaying democracy. Grossly overrated but still good. It has great aesthetics and uses classical music. If there is a flaw, it's that the interesting battle tactics and campaign strategies are frontloaded, and there is way too much standing around and talking about basic philosophy. Adult protagonists. 110 episodes. The new remake series should be seen after
The Vision of Escaflowne: adventure anime set in a Final Fantasy-esque world about rebels fighting against the invading magitek empire, with knights and steampunk mechas and airships. Follows a trio of characters: a teenage prince, a teenage schoolgirl, and an adult knight captain and airship leader. Fantastic aesthetics and directing (Studio Sunrise!) and some exciting fight choreography, and music by Yoko Kanno. Only caveat is that early on, there is a bit too much tarrot card reading vs the war plot, though the latter half of the show is much more focused on the war. The show was supposed to be 36 episodes, but their budget was slashed at the last minute, so after episode 20 when the campaign is kicking up into high gear they had to rush through the final 16 episodes of story in just five more episodes. The first English dub was solid, dunno about the new one. 25 episodes. There is also a movie, which had a boring story but had fantastic visuals and should be watched after the show.
Twelve Kingdoms: a teenage schoolgirl gets isekaied to medieval fantasy China. Focused on geopolitical conflict between the Twelve Kingdoms, but it's mostly political maneuvering and spies rather than massive battles. Good aesthetics and good directing. Soundtrack is okay. Only caveat is that the show is overall slow, especially the first arc.
Gundam 080 War in the Pocket: during a great war between orbital revolutionaries and a corrupt Earth federation, a special forces team of revolutionaries infiltrates an ostensibly "neutral" space colony that is actually hosting a secret weapons project for the Federation. Their mission is to destroy the weapon. Very high production values by Studio Sunrise and likeable characters. Six episodes. A lot of other Gundam stuff is good (08th MS Team, the F91 movie, and Char's Counterattack but you need to have watched the first three shows to understand it).
Death Note: the young adult son of the chief of police stumbles upon a book. Whoever's name is written in the book dies. The son then starts using the book to kill evil people, but then quickly draws the ire of the authorities (including his father) who begin a manhunt for this new remote vigilante. Very high tension. There are deus ex machinas later on but the show was overall really engaging in spite of that. Good directing, good music by Hideki Taniuchi. English dub is okay. 36 episodes.
Neon Genesis Evangelion and the
End of Evangelion movie. Grossly overrated but has great aesthetics and directing.
Sword of the Stranger: awesome samurai action movie animated by Studio Bones. Also has a decent soundtrack.
Dragonar: the first official, unofficial Gundam AU, made by Studio Sunrise and the same people who made the prior Gundam shows and began airing the week after Gundam ZZ finished. It is Gundam (Earth Federation vs rebel space colonies), but with a unique color palette, battles fought by ballistic weaponry (machineguns and missiles) rather than boring pew pew laser spam that plagues almost every Gundam show except IBO, and a trio of three adult protagonists who actually want to be in the military and train and follow orders rather than some bratty teenager who constantly disobeys. The show does get kinda boring in the middle when they're fighting the mad max biker gang. The fan translations are Japanese voices only, no English dub.
5 Centimeters Per Second: a bittersweet movie about lost love by Makoto Shinkai. Young adult protagonist. His other movies are also enjoyable and recommended.
The Place Promised in Our Early Days and
Garden of Words have more of that melancholic feel (also young adult protagonists), while
Your Name,
Weathering with you, and
Suzume are much more upbeat adventure movies with higher production values (teenage protagonists).
Berserk 1997: adapts the best part of the manga, the Golden Age arc. Great aesthetics and good directing, also a memorable soundtrack by Susumu Hirasawa. English dub is okay. Don't watch any other other Berserk anime. The manga has great art but the story falls off after the Golden Age arc.
Clannad and
Clannad After Story. Hard to describe it without delving into spoiler territory. Basically it's about a teenager who grows up into a young man and finds love, and then..
. Starts off as funny school slice of life hijinks but then becomes melancholic as theu get older and life doesn't turn out the way they wanted. Is one of the very few anime where the protagonist gets married during the show, not in the last 5 minutes of an epilogue. Memorable soundtrack.
Almost any Studio Ghibli film. Most of them are at least very good. There are a few ones with boring stories but they are carried by their high production values, even Tales from Earthsea.
Many movies and OVAs, such as Patlabor 1 & 2, Akira, Ghost in the Shell, Venus Wars, Wings of Honneamise, Megazone 21 parts 1 and 2, etc.
Lastly, stuff that isn't very serious throughout like the above, but is good:
Kekkai Sensen: a teenager visits New York City, which now intersects other dimensions and is now populated by aliens and mutants. Crazy shenagains, fun comedy and cool action sequences. Is an adaptation of a manga by Yasuhiro Nightow, aka the author of Trigun. First season (12 episodes) is great, second season (12 episodes) is fine. English dub is fine.
The early
Pokemon movies: fantastic aesthetics and high production values. Movie #3 in particular has a good story that is enjoyable even if you don't follow the TV show. Pokemon kept doing cel animated movies long after everyone switched to digital. IIRC Pokemon Ranger and the Temple of the Sea was the last anime movie that had filmed cels. The movies begin to gradually decline quality. If you only care about aesthetics/production values then you should stop after Destiny Deoxys. By the time of the Lucario movie, they don't look great anymore but the stories are still fun to watch. The Sinnoh trilogy was fun. After the first Black & White movie that they become mediocre. The recent "Power of Us" film was surprisingly good, but lacks the fantasy aesthetics of the early Pokemon films. The English dubs were solid, but supposedly the Japanese audio has better music.
The Big O: Bruce Wayne pilots a giant mecha. Also he has a cute android servant waifu. Was made by Studio Sunrise, so expect great aesthetics and directing. English dub is solid.