As for the recommendations others have made over the last month to new PS4 owners my input is:
Must Buys:
Uncharted: The Nathan Drake Collection
Uncharted 4
The Last of Us Remastered
Bloodborne
Witcher 3 (if you haven't played it elsewhere)
Great games worth playing:
Horizon: Zero Dawn
Games in my backlog I'm looking forward to:
NieR:Automata
Nioh
Tales of Berseria
The Uncharted series is one of my favorite game series of all time. I like story driven games and Uncharted's dialogue, voice acting and story are all A++. I enjoy the gameplay too, especially trying to stealth through entire areas. If you're planning on getting these and also planning on getting The Last of Us I highly recommend playing The Last of Us first. I played all of the Uncharted games before I ever played TLoU and I think it cheapened that experience for me. The game and story are great but I don't hold it in as high regard as many of the posters here (who probably played it when it originally came out in 2013, before having played all these other fantastic story driven games that have released since then).
Witcher 3 is just one of the best games ever made.
I saw a few people recommend Persona 5 and I can see why some people would like it. It's highly stylized and to be fair the combat and Persona system are really good. The options and customization of all the Personas is top notch. However, like I mentioned above, I really like story in my games. I can get past shit story for superb gameplay but Persona 5 is entirely too bogged down by boring, annoying and otherwise tedious systems. I got all the way to December in the game, which is near the very end, and just couldn't force myself to finish. I'd say about 15% of your time spent in the game deals with your Personas and combat and the other 85% is spent simulating a weird, Japanese version of high school life. Not for me. It will likely be the first and last Persona game I ever buy.