I am new to flash photography. I'm shooting a 6D and 580EX II. What is the best shutter to use for general portraits indoors and outdoors? What shutter speed should I not go above?
What is the best shutter to use for general portraits in and out doors?
There isn't one. As with ambient-only exposure, it all depends on your lighting conditions and what effects you're trying to achieve.
You can use the 6D's sync speed (1/180s) as a starting point. It's generally best to stay at sync speed or slower if you want to consider flash power. But if, say, you want to shoot thin DoF outside in brighter sunlight, you may have to use high-speed sync (HSS) to go faster to avoid massive overexposure.
What shutter speed should I not go above?
There's no limit. With HSS, you can go up to your maximum shutter speed. But the faster your shutter speed is, the less power/light your flash can put out.
Keep in mind, ambient exposure is controlled by iso, aperture, and shutter speed; but flash exposure is controlled by iso, aperture, power, and distance.
So your shutter speed only affects the ambient portion of your exposure, not the flash portion. At sync speed and below, the flash burst is faster than your shutter speed, so leaving the shutter open for longer doesn't gather any more light from the flash. That's why you can "drag the shutter."
For basics on flash exposure, I would recommend reading Neil van Niekerk's Tangents website.
Your 6D has a flash sync speed of 1/180. You cannot use faster shutter speeds than that, or else the flash will light only part of your picture.
I would say what is the "best" depends on what you are trying to achieve. If you want to light up the background in a manner that the flash cannot do, you could use a slower shutter speed than 1/180. But do note that if there is fast motion, it can cause motion blur.
If using 1/180 overexposes your image due to bright ambient light, you need an ND filter and also high enough flash power to take into account the effect of the ND filter.