Whenever I pedal, regardless of the gear, I feel a sensation in my soles like I'm turning one of those manual coffee grinders. I can't hear a distinct sound, though.

Turning the pedals backwards, the movement is smooth except for a specific part (left pedal between 5 and 6 o'clock) when it also makes a weird sound.


What could be happening down there? (I'm reluctant to start servicing it)

There's many rotating parts and pieces that could rub if something's out of adjustment. If you're not willing to get in there and get dirty, take it to a bike shop and have them look it over.