The general concept is that the larger the sensor, the shallower the depth of field potential. As you also noted, more zoom or larger aperture can also help to narrow the depth of field, with the three together yielding the maximum ability to control depth of field as needed through sensor size, larger aperture, or more focal.
Your best bet would probably be to move up to a mirrorless camera, which can give you that larger sensor. M4:3 will give you shallower depth of field compared to a small sensor P&S. Move to Samsung NX or Sony NEX, and the larger APS-C sensor can give you even shallower DOF. With the kit lens, you've got a chance when using max zoom, but longer telephoto or faster apertures would be advantageous, even with APS-C sensors. If you're worried about F1.8 being too big an aperture in bright light conditions, you can always use an ND filter to allow you to use the larger aperture.
Some last-year models of Oympus Pens, Sony NEX, and Panasonic GH are on sale in the $350-500 range, and would probably be the ones to look at. The Nikon V1 sensor is quite a bit smaller than the M4:3 sensors, so will be more difficult to produce that shallow depth of field effect without much more telephoto or background distance. The Sony NEX would yield the best shallow DOF effect of the mirrorless set (along with the Samsung NX), and paired with either a fast 50mm prime or a telephoto zoom would be the best bet - the 5n and F3 models are on sale, and even the older C3 might still be around for even less. Any of the M4:3 or APS-C mirrorless would work better than what you've got.
Sony DSLR-A580 / Sony 18-250mm / Minolta 50mm F1.7 / Sigma 30mm F1.4 / Tamron 10-24mm / Tamron 150-600mm / Tamron 90mm F2.8 macro / Minolta 300mm F4 APO
Sony A6000 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses