First, if you haven’t checked if you can get the new GMail interface with categorized tabs, try clicking on the “gear" icon (top right) and click on ‘Configure inbox’ in the drop down (if its there, you can enable this new-ish feature).

I found myself wanting to move between categories with my keyboard.
The trick:

Use the ` key to navigate between ‘Primary’, ‘Social’, etc.

Dishes are done dude.