FiLMiC Pro

For video, my recommendations are Filmic Pro, and Mavis - Pro Camera. You do not need to go paid, but you should look for apps specifically designed for the task. In my opinion, Filmic and Mavis are the best, and I don’t think they cost all that much in comparison to other things you’ll need to get going. That said, you are more than capable of getting by using most stock camera apps.

I'd strongly recommend you invest in Filmic Pro for iPhone shooting. It gives you far more control over framerate, resolution, and bitrate than the built-in app as well as additional professional-grade features. Pretty much all the 'shot on iphone' Apple get made are shot on Filmic.

I tried a few apps and Moment or Filmic Pro both give excellent results. You can shoot in RAW, ProRaw, or JPEG and you have the option to turn Apple’s photo processing algorithms off or just lower their effect on your photos. Of course, the best feature of the third party apps is manual focus and the ability to separate focus and exposure points. That alone results in much, much more freedom when composing your shots.