Ah, grasshopper - you have opened a question that has many answers. And no answers.
Of course, you can always use 'it depends'.
24 fps is the 100 year old standard for film / cinema look. People are used to it and the various attempts to change it (i.e., Peter Jackson's 'The Hobbit' at 48 fps) haven't met with much success. 30 FPS is standard for television and so has, mostly by default, become standard for computer video (actually is 29.997 but we will ignore that for now).
At the slower speeds, the image of course blurs when you move it. Again, we've grown up thinking this is normal and expected and film makers have a long history of using this to convey motion or other effects. Higher frame speeds pick up more detail. This may be useful for scientific / engineering / non artistic purposes but many people feel that it looks 'jittery' for most scenes. But if you're trying to capture snowboarders, it makes lots of sense.
60 FPS is designed for fast motion. Unfortunately, DJI kinda botched the implementation and it looks pretty blotchy and has lots of artifacts. Perhaps they will fix this but it's been noticed since they first came out with that speed so don't hold your breath.
For most people, 30 FPS is probably the right place to start. But you should understand what you are doing.
fasted's contention that 1080/60 'simply looks better' would not be a popular statement. YMMV and of course, there is no accounting for taste. Perhaps for his particular type of shooting it works best but I would not just park my camera there. I use 4K 24 fps because I do mostly nature scenes with slow pans and transitions. When I try to capture eagles running around I use 1080/60 and curse DJI's engineers.
This video explains it nicely.
In terms of capture size, unless you need the 60 FPS and assuming you have a computer that can handle it, shooting 4K makes the most sense. It helps future proof your images - have you taken a look at the 640x480 stuff you did years ago? VHS? It also lets you crop the image which can be very useful. The 4K implementation in Phantoms isn't first rate by any means - too limited a bit depth (another technical discussion) and limited hardware (a bit too slow). But hell, it's only $1000. If you want pro stuff you need to add one or two zeros to the right side....