Well, I'm a mature, manly individual, so I watch (re-watch, in fact) RWBY. It's awesome!
I was getting into anime about 20 years ago ... and then I got married. I watched The Vision of Escaflowne back then and blew my mind, wow! Would be cool to take a dip back into it, but there's so much out there I wouldn't know where to start. I'm open to a road map ... hint hint 
While RWBY is marketed as an anime, it's actually an American production, and distinctly low-budget one at that. In no way, though, does that take anything away from the show. I particularly like how it improves with each subsequent season. The only thing is, it's fairly simplistic, more so in the beginning.
Considering anime in general, I'm somewhat versed in the subject, but at the same time reserved about it. In essence, anime falls completely within the purview of Sturgeon's law, thus being not what exalted young people would lead you to believe it is. Furthermore, I would say it's very hard for mature audiences to get into anime, since it's either very strange or childish. Me, I like childish stuff, so there is no problem. But it's not for everyone.
All of this is compounded by the fact that many great anime shows are completely genre works, requiring you to watch many of their predecessors. It's the case with Puella Magi Madoka Magica (magical girls, with Sailor Moon as the trope codifier) and Kill la Kill (80s-style fighting animes). I think both are interesting to watch, with Kill la Kill being amazing and funny in particular, but I don't think it's possible to enjoy them outside of context.
That being said, my hands down favorite anime (manga, to be precise, but anime follows it almost word for word, fillers notwithstanding) is One Piece. It has, so far, only around 800 episodes. It's not even close to being over.
An anime show to start out and get a somewhat relevant look at the medium is, perhaps, Fullmetal Alchemist. More specifically, Fullmetal Alchemist: Brotherhood, because it's a complete and concise story lifted entirely from the original manga (it's a good thing, and I wish it were done more in the West).