If you’re looking for something modern, convenient and efficient – the Sony Cybershot and Nikon Coolpix range have some nice compact cameras. I’m sure all manufacturers have nice compact cameras, but these are the ones I have used specifically and would recommend. The most important thing to look for in a camera is the image sensor. Pay no mind to mexapixels – unless there is an insanely high amount, because this usually means all the data is crammed into a tiny image sensor. Six mexapixels is more than enough in a compact point and shoot camera.
@jambon_777 You specifically asked for a digital camera, but have you considered a film camera? In comparison to digital point and shoot cameras, film cameras are superior in dynamic range, resolution, and tonal quality, amongst a myriad of other things. 35mm SLRs are the easiest to learn, however if you’d like something a little more compact a rangefinder would be sufficient. Both of these can be had for well under $300 – usually well under $100. You mentioned that you didn’t want to carry around a large lense, so I’d suggest getting a prime! 50mm lenses are super cheap and easy to come by in any old camera shops, and because they usually have “fast glass” – they perform exceptionally well in low-lighting without the aid of a flash.