Most of my exposure with anime was whatever was on Toonami and Adult Swim in the 2000s. I've not watched stuff like, Ghost In The Shell, Cowboy Bebop (only disjointed episodes out of order), Neon Genesis Evangelion, which I know are highly regarded.
I've seen just about every Miyazaki movie at this point. I've seen Akeria.
I'm sure there's more mainstream shows and non mainstream shows that I'm missing and would enjoy.
Thoughts?
1997 - Revolutionary Girl Utena. Arthouse feminist show by one of the directors of Sailor Moon. Also has a movie.
1998 - Cowboy Bebop. Atmospheric classic inspired by western cinema and beloved in the west to this day.
1998 - Initial D First Stage. Exciting racing show with an incredible soundtrack. Season 1 might have the worst CGI, but that's also where the show peaks, as the following seasons have a worsening pacing and side characters get pushed out of the frame completely by the dreadful Season 5.
1998 - Serial Experiments Lain. You've probably heard of this one.
1999 - Turn A Gundam. Yoshiyuki Tomino's late career classic.
1999 - [CW] Great Teacher Onizuka. This is a show of contrasts. Where it's good, it's really good. Where it's bad, it's really bad and uncomfortable. It is however absolutely a show that was considered a classic for many years. One of the characters is treated in a very ableistic way unfortunately.
1999 - The Big O. Again inspired by western media, it was a huge flop in Japan - but as a mecha show it does what it does really well.
2002 - Azumanga Daioh. The first slice of life/random nonsense go show, and at the same time the best at it - rivaled only by the more absurdist approach of Nichijou.
2006 - Welcome to the NHK. A show about social isolation, being preyed upon by capitalism and mental health issues. Has some flaws, but is absolutely worth watching.
2007 - Kaiji, Ultimate Survivor. A gambling drama that is highly likely hook you really hard, sprinkled with anti-capitalism on top and Nobuyuki Fukumoto's unique signature artstyle.
2009 - K-On! The platonic ideal of the pure slice of life show. IMO it only gets good in season 2.
2011 - Nichijou. As mentioned before, a more surreal Azumanga Daioh, with quality animation by the workers of the renowned Kyoto Animation.
2012 - Jojo's Bizzarre Adventures. Similarly to Breaking Bad, it lives up to the hype.