I second Killzone if you want a little story to go with your shooter. You aren't going to get The Last of Us style storytelling out of it but there is certainly more there than the Call of Duty Ghosts or Battlefield 4 campaign. If you are a fan of those series' online play you will probably feel right at home in the PS4 versions. I own all 3 of those games. I have spent quite a bit of time with them online and they are all different experiences but fun in their own way. Your personal tastes would dictate which one you would like best. I would choose Killzone for single player campaign by a landslide and then probably Battlefield 4 for online multiplayer but that one is by a closer margin. You should be aware that Battlefield has some glitches in it's online play if that really turns you off.
The game I have actually spent the most time with on my PS4 is Assassin's Creed IV: Black Flag. That's a little strange because I wasn't really a fan of the earlier games in the series but I liked this one a lot. If you want to read my much more in depth thoughts on it you can read this post.
http://www.satelliteguys.us/threads...tellite-Guy-s-gamers-at?p=3370342#post3370342
I'm going to assume you have PS Plus because you need it for online multiplayer on the PS4. If you do have it be sure to download your free copies of Resogun and Dead Nation. These are smaller games but I actually played Resogun long enough to get the platinum trophy so you might like them. Make sure you check the online store and "purchase" all the free games at least once a month even if you don't think you will ever play them. You don't have to actually download them but once you "purchase" them they will be tied to your account even after Sony changes the list of free games out on the online store each month. Outlast and Don't Starve just got pulled down so you already missed out on those freebies.
https://store.sonyentertainmentnetwork.com/#!/en-us/free-games/cid=STORE-MSF77008-PSPLUSFREEGAMES
There are also a couple free-to-play shooters that are both pretty well received. Warframe is a third person co-op shooter where the combat looks kind of like Mass Effect. Blacklight Retribution is more of a standard, online versus shooter. You can get into both of these for free. I haven't spent any time with either of them but I have them both downloaded. The free-to-play model turns me off a little but lots of people love it.
Edit: I forgot to mention inFamous Second Son is only a couple weeks away at this point. This is the big exclusive most PS4 gamers are waiting for. The reviews aren't out yet so this could end up being a big disappointment but the general buzz seems to be positive so far.