It's going to be darn hard to "de-focus" too much using a digital camera while keeping the main subject in sharp focus. The smaller the sensor, the greater depth of field you will have. Generally speaking, the smaller the camera, the smaller the sensor. Your Panasonic DMC FZ8 has the smallest commonly used sensor (1/2.5"). Most of the pictures you take with a digital camera are quite sharp from near to far distances and there is a reason for that which I will explain.
While we speak in terms of the 35 mm equivalency of digital lenses, don't forget that the digital sensors are usually smaller than a full-format 35 mm frame. Most of the more popular point and shoot cameras (including your Panasonic DMC FZ8) have the smaller sensors. It's only about 5 mm wide and 4 mm high. The lens on required to cover that angle of view is an ACTUAL 6-17 mm zoom lens. At these focal lengths, the background is going to almost always be in pretty sharp focus.
In other words, if you WANT to defocus the background, you are going to have to work pretty hard at it. You would have to zoom to the longer end of the lens and set the aperture open as wide as it will go, if your camera even allows you to control the aperture, and get pretty close to your main subject while having the background a fair distance away. Since you have a "superzoom" camera, you have the advantage of some longer focal lenghts available to you.
Put your subjects quite a distance in front of any background that will be visible in your photo. You could use a garden of flowers and stand your subjects 20 yards or so in front of them. As suggested, try the portrait mode to shift things towards a larger aperture, zoom the lens out all the way, and move YOURSELF backwards or forwards in order to frame the picture as you would like it.
This is why the first shot (the chrome mirror housing) worked. You used a much longer focal length in this image than you did for the row of cars. To do the same with the row of cars, you would have to walk about another 30 yards or more away from the point where you were standing and zoomout enough to get the same composition. This would allow you to blur the more distant cars nicely, as you see in the first shot. Another thing that is happening in the first shot is that you have focused on an object that is very close to the camera. The longer the telephoto and the closer the main subject, the more the background will become blurred.
The bigger the sensor, the easier it will be to achieve pleasing bokeh. This means moving to a dSLR, which all have sensors about 20 times bigger than the typical P&S digicam. If you want to really go for brokeh (very bad pun intended), you can get a Canon 5D and you will get exactly the same effect you are accustomed to in a 35 mm camera, since the sensor is the same size as 35 mm film.
Here is an example with a point and shoot camera, although it does have the larger sensor (1/1.8") that makes it easier to blur the background. Even though this is macro mode and f/2.8, where the background should blur the most, it's not terribly blurry because it's relatively close to the subject.
http://www.flickr.com/photos/samfeinstein/872732755/
Here is an example with a point and shoot camera, so it CAN be done. The background is much farther away, though, and this is the larger sensor size.
http://www.flickr.com/photos/samfeinstein/843563558/
Compare that to this image, though, which has a similar subject-to-background distance. The SLR has the obvious advantage.
http://www.flickr.com/photos/samfeinstein/408446616/
Wikipedia does pretty well on the subject of depth of field. See: http://en.wikipedia.org/wiki/Depth_of_field
See also: http://www.photo.net/learn/optics/dofdigital/