Also not a fan of FPS shooters, though even they take up most of the market I have managed to find a good amount of fun games that do not fall under that category.
You can always just stick with the free games the PSN gives you monthly (I assume you'll get + if you want to play games online like Overwatch) You can also check out their sales at their online store when they have them (you can access them on the PS4), sometimes they drop the prices really cheep.
If you want story driven third person action game the Uncharted series is a good one (you can get the collection which is the first 3 games for the price of one, and then there is 4) I really enjoyed those ones, 2 and 4 being my favourites.
Last of Us is also pretty good from what I played, though I still have to finish up that one. Similar style to Uncharted, but not exactly.
Depending on how long it's been since you played a FF game, there's the remastered versions of X and X-2 and a port of Type-0. Those titles are both relatively cheep by now.
Disgaea 5, I really love this series, personally I don't think 5 was as good as 4 but I still enjoyed my time on it. Tactics RPG that likes to make fun of itself, level cap 9999. I invest way too much time in this series.
Ratchet and Clank was a lot of fun, and it looks great on the PS4. Platformer game, very cute.
Odin Sphere Leifthrasir Seriously, I can't recommend this one enough, it's a remaster from the PS2 and did they every perfect what I would already consider a gem on the ps2. Beautiful 2D Side scroller / RPG.
For indi titles Journey, Don't Starve (Don't Starve Together if you want the multiplayer version) Armello, Binding of Issac, Bastion, Transistor, Geometry Wars, Limbo are some that come to mind, I'd have to check my library... PSN+ gives me so much crap I never have time to play it all.
Infamous isn't bad from what I played though again, another titles I never got around to finishing. And I think there is a God of War something remastered for the PS4, enjoyed those games on the PS2/PS3 though fair warning, they do a lot to deserve the M title, so if over the top blood shed and nudity is not your thing stay away.