Arguably the best app for shooting video is FiLMiC Pro. Versions are available for Android and iPhone.
I don't know much about Android camera apps, but for processing I always recommend using Snapseed. It's fast, easy, and can do most of what you might want to do.
For getting photos onto a PC or Mac, I recommend PhotoSync. It can be faster than plugging your phone into the computer, waiting for it to show up in Explorer or Finder, then navigating to the photos, selecting them, and moving them. In fact, once it's running, all you have to do is select your photos and move them. But it can do far more than that such as creating subfolders on the fly, deleting the originals, and a lot more. It's also usually faster than using AirDrop. There's a paid version and a free version, the main difference being that the free version now transfers photos in JPEG at a lower quality, although you don't need the paid version on the receiving end. The paid version is only £13.50, which is worth it if you use it after every shoot, as I do. PhotoSync is available on Android, iPhone, PC, and Mac.
Since I use iPhone, the rest of this is for iPhone users, so thanks for making this an open thread.
For processing video, Lumafusion is reckoned to be the best. I use it and it makes everything I want to do so easy. Occasionally I have to go back to Vegas on the PC, e.g. if I want to use masking. There quite possibly won't ever be an Android version, though I wouldn't rule it out. People also want Mac and PC versions, but so far nothing has been forthcoming. Lumafusion is easy enough to use on an iPhone, but I recommend using an iPad.
For shooting photos, I can recommend FiLMiC's new camera app, FirstLight. It gives you more control over the camera than the built-in Camera app and also includes some film-style filters, with more available if you take up a subscription. ProCamera is also good, and has a cheaper subscription model.
For processing photos on the iPhone, I'll always recommend Snapseed first as it's so easy to use, it's fast, and it does most of what you might want to do when it comes to making both global and local adjustments to a photo. Its main drawback is that you have to run Snapseed then open a photo - you won't find it (or be able to put it) on the Share sheet. As a good Lightroom substitute, Affinity Photo can't be beaten. Other good options include Lightroom Mobile, which also allows you to set Exif data while the others don't; and PixelMator which is a good rival to Snapseed.