Out of all of my digicams, the ones I love the most are the Sony Cyber-shot DSC-W570 from 2010 and the Nikon Coolpix 3200 from 2005. The nikon has that very old school camera quality to it, but xan be quite bulky to carry around, its SD card slot caps at 2GB (and is not compatible with sd-hc cards, normal sd cards may be a bit difficult to find now) and can take a bit to save pictures after taking them, making it horrible for capturing fast moments (moving objects, fireworks, etc). The sony on the other hand, is a yaw more lightweight and sleek camera, perfect to carry around. It captures images very fast and is compatible with sd-hc cards. Its images tend to look a bit warmer though (depending on the lighting, but it has more tendency to capture oranges undertones compared to the nikon). Both these cameras are pretty old, so they should be pretty cheap on the 2nd hand market.
It wouldn't be fair to suggest any digicams without showing pics that were taken using them, so here are a few :)
Nikon Coolpix 3200
Sony Cyber-shot DSC-W570
I hope this helps :D