I always recommend to those who never watched anime or just think they are just silly cartoons with big eyes characters – Death Note. An easily understandable and original plot and yet a lot of twists, turns and suspense. Other is Code Geass – a mecha aspect can be a bit distracting but story and characters are as good as in Death Note. And third is Monster – a psychological thriller, a bit slow in the beginning, but then it becomes very suspenseful. Attack on Titan – still on-going series, but should be ending soon, wont go into it much, because its probably on everyone’s ear right now, it’s good, plot is pretty original.
If someone likes good cyberpunk I recommend Ghost in the Shell: Stand Alone Complex 1st and 2nd Gig and Ergo Proxy (this one is a bit complicated so maybe not for everyone, probably better not take a long breaks between watching episodes).
If someone into romance mixed with other stuff: Your Lie in April (a good slice of life story with great soundtrack), Steins;Gate (this one has time travel, so if you are into that), Chobits (this is one is also funny) and Nodame Cantabile (this one is with classical music).
If someone into episodic anime with just good stories I recommend Natsume’s Book Of Friends and Muhishi.
Movies made by Makoto Shinkai are very good like Your name and 5 centimetres per Second. 2 very first Ghost in the Shell movies too.
Of course there is a lot more, but these should be the good ones to start with.