There's quite a lot of different photographic flaws that you can test for. For each of these tests you may get different results at the widest angle compared to the longest zoom.

1. Autofocus - Try setting up a high-contrast subject like a black & white test card and use autofocus from different distances while the camera is on a tripod to take shots. Check the sharpness of the results.
2. Vignetting - Use a plain card under even lighting and check how much difference there is between the light center and darker corners.
3. Distortion - take a straight-on shot of some graph paper, and measure how distorted the lines are at the edges or in the middle.
4. Auto White Balance - take shots of white paper in different lighting conditions (sunlight, cloudy skies, normal indoor lighting, tungsten lighting) and check how close it gets to white
5. Representation of colour - take shots of a standard test card which has both brightly saturated colours and flesh tones, and compare the results with what you like

There's also different capabilities that might restrict the choice of camera - e.g. macro, depth of field, flash, manual focus, etc.

Ultimately, testing this stuff yourself is interesting but might be hard work. For comparing specifications, there are a few different online solutions. For comparing test card type images there are fewer options as taking consistent shots is very difficult.