Understanding Export Quality Settings in Lightroom Classic

"I'm having a hard time understanding if I should let the image sizing box do its own thing or if I should manually enter 300dpi for every photo I want to export for printing" My understanding is that the RESOLUTION option only comes into play when you are selecting a physical dimension in the export. For example, in the  following, I select 8x10 inches.

The resulting image is 2400 (10x240) x 1690 (240x8=1920 but it's cropped) pixels. 

... or if I should manually enter 300dpi for every photo I want to export for printing.

You can't enter 300dpi (dot per inch) in the Lightroom Classic Export dialog box. You could enter 300ppi (pixels per inch) but there are two extremely different things — dot per inch meaning how many dots of ink a printer will use per inch, and pixels per inch, meaning how many pixels of the photo get printed per inch. These both are entirely meaningless if you are not going to print anything. They do not affect digital images. They only affect printed images. And if you are going to print something, you can only control ppi from Lightroom Classic.

That 300PPI is kind of meaningless as it's just a resolution tag. The number of pixels is key here. Work in pixels. 

1000x1000 pixels at 1000dpi and 1000x1000 pixels at 10dpi are the same. See this old article that to this day still applies:

http://digitaldog.net/files/Resolution.pdf

Then in terms of ideal resolution for some print work, see:

https://www.digitalphotopro.com/technique/photography-workflow/the-right-resolution/