Panasonic G3 + P100-300
The first picture can't have been motion blur because you were at 1/2000 second! Most likely a focus issue. With a subject like this it is best to use the smallest single area focus box and pick out the eyes. Pinpoint focus is good too as long as there is no subject movement.
The 100-300 at the long end is best stopped down to F/7.1. F/8 is good too but not any better so you might as well let as much light in as possible. Most of the time, with OIS on, 1/500th second is enough so set to Aperture priority and F/7.1 and you're good to go. Allow the OIS to settle for a second or so before taking the shot. The viewfinder image will go smooth once this happens.
If your subject is moving and you want to use a faster shutter speed, things are a bit trickier because there is no Auto-Iso in manual mode. So, to get F/7.1 and say 1/1000th second, set to Aperture priority and F/7.1 and then set manual ISo to give you roughly the shutter speed you need. This way, the shutter speed may vary a bit around your target of 1/1000th but at least your exposure will be set automatically.
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