The problem with recommending anime is that its such a unique medium from "normal" TV/movies that it is its own genre. Unless you're prepared to dive in head-first and embrace whatever you find "cringe", you likely will be turned off by the very next one you watch. Is it the absurdism that turns you off? Because that would shorten the list down considerably.
In addition, the fact that there are so many different styles and sub-genres spanning the gambit from fantasy to hard sci-fi, romance to tragedy, slice-of-life to horror, realistic to surreal, that unless you can give specifics you'll probably just get the usual reel of "you have to watch
this or
that" with no context or real direction for what you're actually wanting. I can only speak for my tastes, which are quite antiquated compared to most these days, but even in the backlog there's plenty to find.
That said,
Cowboy Bebop is generally a good entry point, as some others have said. I'll try to recommend those that haven't already been mentioned, but Bebop is such a cornerstone it bears repeating.
Summer Wars - a story about a girl trying to simultaneously deal with the loss of her grandmother, her diverse and rambunctious family (who are around for the funeral and as a general family gathering), and stopping a doomsday online virus threatening the world at large - is a single movie that you can plow through in one go.
If blood and gore are more your pace,
Hellsing Ultimate is pretty much king even now, both in terms of a decent story to follow, and having Tarantino levels of viscera sprayed and exploded every which way.
The
Ghost in the Shell movie can be a bit shallow at points when its trying to be philosophical, but the action is excellent and effects/visuals stunning even now, and the questions it raises when it hits the mark story-wise tend to linger. I haven't watched Stand Alone Complex or its sequel series (although I intend to eventually), but have heard great things about them.
Gurren Lagann is essentially the same coming-of-age story as NGE, with all the angst thrown out the window and blasted into dust in favor of truly absurd mecha fights. If that sounds like your thing, there it is.
The first OVA episode of
Dominion Tank Police and the entirety of
New Dominion Tank Police are insane dark-ish comedies about a police force that uses tanks to deal with the city's criminals. Primarily the pissed-off valkyrie bitch that builds her own (mini)tank and saves the day while everyone else makes a mess of the city. Plus both OVA and the series have
absolute bangin' openers.
For a lesser-known cult classic, I present
Haiban Renmei, a slice-of-life about girls with no past who appear to be reborn in a new world, and must come to terms with their lives before they are taken away. A lot of drama, but very well paced, and just enough world-building is given to let us see what is happening, but not enough to spoil everything all at once.
I can't have any of these recommendations without mentioning
Outlaw Star. Excellent and timeless tale of a crew of entertaining misfits looking for the Galactic MacGuffin (not the actual name, but essentially what it is
), and excellently animated by Sunrise with a fantastic OP.
And right alongside that is the amazing Western-themed
Trigun, which mixes comedy and tragedy almost perfectly, and is one of the rare instances of the anime arguably ending better than the manga it was based off of. Vash, while initially coming off as a goof, remains amongst the most complex main characters in all of anime, and his supporting cast & rogues gallery are both top-notch. Add to that fantastic animation and a great soundtrack that mixes country twangs with a bit of modern tastes, and its just all-around a great watch.
Finally, I have to mention
Jin-Roh: The Wolf Brigade, a complicated and somber story of a post-WW2 alt-history Japan, and the heart-rending story of a young recruit into the titular Wolf Brigade who falls in love with the sister of a resistance member. Its not for everyone, and not an easy watch for the weak of stomach, but remains among the best anime movies of all time, hands down.
