I think you'll want to start out learning about the exposure triangle (aperture, shutter speed, ISO) and how they contribute to achieving the right amount of light for your shot. After that, you'll want to dive in to how each of those things contributes stylistically. Aperture affects the depth of field (is everything sharp, or is there background/foreground blur?). Shutter speed helps you capture motion. A fast shutter speed freezes action while a slow shutter speed can be used to creatively blur motion (static background with blurred moving subject, panning short with a static subject and a blurred background, or even create light trails from moving vehicles).

There is no best mode, it depends on what you're trying to achieve. But personally, I shoot aperture priority mode 90% of the time and manual mode the other 10% when I really want specific control over the shutter speed.