Based on a game by Colors.

What is the best anime for a newcomer to watch?

I fully agree with the recommendations of Death Note and Fullmetal Alchemist Brotherhood made by other quora users.
Here is a few more I'll add. My Hero Academia Easily the best mainstream shounen anime of the decade, it's incredibly accessible for an English speaking audience, plus on several streaming sites.
The story is about a boy named Izuku “Deku” Midoriya is a superhero society trying to achieve his dream of becoming a hero.
It has great chararcters, satisfying fights, and an addicting story.
Plus (ultra) it's still airing so you won't be too far behind when it inevitably becomes a juggernaut of a series to get into. Spirited Away By far the best anime movie to start with, it tells the magical tale of a young girl who finds herself stuck in the world of the spirits.
It is the only anime to ever win the Academy Award for best animated picture, and it is one of Hayao Miyazaki’s (a name you will become familiar with if you pursue anime further) best films, besides his masterpiece Princess Mononoke.
Only reason that isn't on the list instead of Spirited Away is it's not the best movie to start with. Your name. The highest grossing anime movie of all time, this is the other great movie to introduce yourself to anime.
It is an absolutely gorgeous visual masterpiece about two teenagers who suddnly find themselves switching bodies.
It is an incredibly emotional tale that has received incredible critical praise.
It is currently ranked as the second best anime of all time on a site called MyAnimeList, which is a great source to find new anime to watch. Cheers!