If you want the unfocused background with a P&S camera like your S6000, the keys would be:
Set the camera to the 'portrait' or 'macro' scene mode - Usually something like a flower or silhouette of a person.
Stand farther back from your subject then zoom all the way in until you've got them framed how you want. Watch the aperture - you want the smallest F number the camera can attain (F5.6 at full zoom)...if it's higher than that, make sure the ISO is set manually to the lowest setting. Keep as much distance between your subject and the background as possible.
It can be done - it just takes more work and thought. These were done with a tiny sensor P&S zoom camera:
It's possible - it just takes a little more work, and a little help with distances.