It's EXTREMELY difficult to use a point & shoot camera and create beautiful out-of-focus background blur.
"Bokeh" refers to the quality of the blur (and not necessarily the strength of it). But in photography, we can create an out-of-focus blur by having a subject at close focusing distance, a background which is far beyond the subject, using a lens with a long focal length, and using a low focal ratio. All of those properties contribute to the strength of the blur, but the quality of the blur is largely a property of the shape of the lens aperture (how many aperture blades does the lens have and are they well-rounded?) and the quality of the glass.
If you used a DLSR... Say a Nikon D3300, and you used a lens with a slightly long-ish focal length and low focal ratio such as a Nikkor 50mm f/1.4 lens (or an 85mm f/1.4 would be even better!)
Here are examples of Nikon's 50mm f/1.4 glass:
Full-size sample photos from Nikon 50mm F 1.4
The reason you struggle to do this with your Coolpix camera is because the lens is a 4.5-63mm zoom lens (ignore the 35mm "equivalent" size, you have to use true focal length to calculate depth of field). But at that 63mm focal length (and frankly even at the 50mm focal length) the aperture is f/5.9 -- which isn't very low. Then add to this that the sensor size is very tiny (11mm diagonal) vs. a Nikon DSLR which is much larger (over 28mm diagonally -- a factor of about 2.6 times bigger) and you have to multiple the sensor crop factor by the focal ratio... so f/5.9 becomes f/15.34. That would be like a DSLR shooting the same shot with a 50mm lens at f/16 -- which would have such broad depth of field that most things would be in focus and you'd get almost no out-of-focus blur (what little you would have would be extremely weak.)
To get the effect you want, I'd suggest looking at a DSLR and then pickup a low cost 50mm lens to get started.
A Nikon D3300 will do it with a Nikkor AF-S 50mm f/1.8G lens... see:
Full-size sample photos from Nikon 50mm F 1.8
A Canon T5 will do it with a Canon EF 50mm f/1.8 STM lens... see:
Full-size sample photos from Canon 50mm F 1.8
(Note that Canon just replaced the EF 50mm f/1.8 II with a new model called the EF 50mm f/1.8 STM. The new "STM" version has 7 aperture blades instead of 5 and a better/quieter focusing motor, but the optics are otherwise the same. The aperture blade change causes a big improvement in the quality of the out-of-focus bur. The 5-bladed version had a "jittery" quality to the out-of-focus blur. The Nikon 50mm already had a 7 blade aperture (the same as the new Canon lens).
I picked the cameras above because they are at the low-end of the price range on DSLR cameras but what you want doesn't require a high-cost camera "body" -- it's more about the camera "lens" (but you do need a body with a large sensor and all entry-level DSLR cameras have a large enough sensor to do what you want ... as long as it's paired with the right lens.)
I picked the 50mm f/1.8 lenses above because they can do what you want at a reasonable price-point. It's probably the least expensive way to get the effect that you want.
For as good as these examples are with 50mm f/1.8 lenses... the effect is even stronger with 50mm f/1.4 lenses (but those are more expensive). An 85mm f/1.4 lens would be even better yet, but you'll notice the price tags really start to rise as you get into the better quality lenses.
I have a Canon EF 135mm f/2L lens that can generate such a sweet & creamy bokeh quality that just looking at the images will give you Type II diabetes and heart-disease all at the same time. ;-)
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