You're asking a few different questions here ... some background on why you use different frame rates :
(although this is a US-centric resource, so leaves out any discussion of PAL/25fps) ...
.... also be aware that iPhones often shoot in variable frame rates, which is problematic to edit
fps will not affect quality, generally speaking.
However, fps controls motion. in other words, a higher fps allows you for a better slow mo.
Also a higher fps retains more motion details, similar to shutter speed (i.e: Rain will be more defined with a higher fps, allowing less motion blur and a more crisp droplets)
As for the look, lower fps is considered more cinematic, higher fps gives an amateurish look and feel.
Best practive is to film with as many higher fps as light allows, allowing more slow mo capabilities, and exporting with 24fps giving a cinematic look.
Hint: files are usually larger with a higher fps, also some cameras are not capable of combining a high fps with a high resolution, so its always better to assess your needs. for example, if you are filming an interview, it is best to film with 24fps, allowing a higher resolution with some cameras.
I exported the same Premiere project (which was filmed in 59.963fps) twice, both in H.264 format. One was at 59.963fps and the other at 23.976 fps. Quicktime property inspector tells me the 2nd is actually 23.98fps — not sure why.
Compared to the 59.963fps version, the 23.98fps version is a little choppy. Not horrible but noticable side by side. Certainly not a better, more 'cinematic' feeling.
Any idea why exporting a 59.963fps file to 23.976 fps in Premiere gets choppy?
You ask 'will there be loss of quality especially with panning and moving shots' at lower frame rates.
Yes, but it depends.
Certain panning speeds in particular will look different and likely 'juddery' at a lower frame rate BOTH converting 60fps to 24fps AND if you shoot at 24fps*. And how it looks is also affected by the shutter speed on the camera. Drones/gopros etc. often have high camera shutter speeds and therefore don't have much 'motion blur' on movement within a shot. It's this lack of motion blur that will contribute to how 'juddery/choppy' it looks when you output (or shoot) at a lower frame rate than what you currently do.
I would suggest staying at 60fps or output at 30fps rather than 24fps.
A rule of thumb would be to output at the frame rate that *most* of your footage was shot at.
Though there are complex other reasons why the above would not apply i.e. creating a program for broadcast etc.
Ultimately output what looks and works best in your use case.
* for the sake of easier typing I'm using 'rounded' frame rates.