I hope I don't get blasted for this post as I know there are countless recommendation threads and blogs about anime all over the place. This seems like an active OT so I figured there would be some good opinion here.
So to my question, I'm a complete anime newbie. I've always wanted to check out some anime but the amount of content out there has always been pretty overwhelming to me. I've watched about 1/3 of Cowboy Bebop, which is pretty much the full extent of my anime viewing history.
I downloaded the Crunchyroll app on my PS4 and found there is a ton of content available on there. Are there any new/currently running shows on Crunchyroll that are good and would be recommended viewing for a complete anime newbie?
Hmm. The best show currently on Crunchyroll is Shirobako, but that's kind of inside baseball, so I wouldn't recommend it for a newbie. That's a show you work towards watching after you have some experience under your belt.
Without knowing what genres in particular you like, I'll just recommend a few high quality shows from the last few years on Crunchyroll.
From the New World is an excellent science-fiction story, with a real emphasis on strong world building and social commentary. It's got some decent action in places and some romance and character development, but the real heart of it is just the way that it builds and fleshes out a society.
Space Brothers is an excellent feel good show about going to space and following your dreams. It's got an entirely adult cast (which can be helpful compared to the often very young casts found in most anime) and spends almost all of its time just developing the characters and having them work towards heading to space. It's a show that will almost always make you feel happy while watching it. At 99 episodes it's fairly long, but it keeps up a pretty consistent level of quality across the whole series.
Flowers of Evil is a pretty different series, and the visual style isn't for everyone (the characters are all rotoscoped, so they can look pretty strange), but it's one of the most enthralling dramas I've ever watched. It's a show that really just highlights what complete shits middle school aged kids are, and just about every episode had me feeling like I was going to drown while watching it. It's a lovely and very atmospheric show that will just really up the sense of despair at every turn.
Fate/Zero is a pretty solid action series with some enjoyable characters to watch. If you're looking for fun action, you could do a lot worse than this show. It's not perfect, but it's a pretty easy show to get into for newcomers.
If there are any specific genres you're really interested, I'm sure people could come up with more specific recommendations. Anime is a pretty broad medium, and there are stories in pretty much every genre imaginable.