Some people assume because I've taken quite a few photos that I'm an expert on photography. It is true that I taught both O level and A level Photography but that was nearly twenty years ago. I also ran a Photography Night School class but even that ended well before the advent of digital cameras. Still I have owned quite a few cameras including six digital cameras so I suppose I possibly know more than a lot of people. The question people mostly ask me is "what camera should I buy".
Well first of all it depends what you want to use it for and secondly how much you want to spend. I have two cameras; a simple point and shoot Fujifilm A500 which cost me 65€ and a Canon EOS 350D with two zoom lenses and an external flashgun which cost me considerably more. I take the compact out with me mostly everywhere just to get the odd snapshot and the results are very good but if I am going out to take some serious pictures I'll take the Canon.
These days there are cameras to suit every type of user at all price points. For most, one of the compact cameras is ideal because they are small, versatile and inexpensive. The more you pay - the more features you get. Some of these are useful, others you will rarely use. I would guess that most people (even experienced photographers) use their cameras in Auto mode for at least 90% of their pictures. So having lots of different settings may not be as much of an advantage as you think. If you do get serious though it is good to have manual control of exposure, shutter speed and aperture.
A decent zoom range is useful but if it is a long zoom make sure the camera has image stabilisation otherwise your telephoto shots will end up blurred by camera shake. Like most people, you'll probably use the wide angle end of the zoom more than the telephoto so make sure it goes as wide as you need. Don't be fooled by digital zooms, all they do is enlarge a part of the image which is something you can do later on your computer anyway. The other major thing to be aware of is that the built in flash on any camera is only useful up to about 3m or so. If you plan to take lots of indoor pictures get a camera which will take an external flashgun.
Call me old fashioned but I prefer cameras built by camera manufacturers. I know Panasonic, Casio and the like make good cameras and even though Sony make the sensors that others use, I would still prefer something from companies like Canon or Nikon.
The plague of many compacts is digital noise. The small sensors that are used, especially those packed with plenty of megapixels, combined with small aperture lenses, leads to noisy pictures when the light is less than ideal. The other problem, especially with the cheaper cameras, is the speed at which they work. My Fuji is slow to start up and slow to write images to the memory card. Potentially I could loose a lot of good shots because of those seconds of delay. If you are intent on taking lots of action shots or you want to take shots in low level natural light you should seriously consider a DSLR (digital single lens reflex) rather than a compact.
The maxim is to try before you buy and do your research. If the camera feels too small or too large and clumsy or the buttons too fiddly then is not the right tool for you.
As you are reading this on the Internet, you have the perfect source for finding out all you need to know about your intended purchase. Go to http://www.dpreview.com/reviews/specs.asp for a wealth of information about every digital camera on the market including those that you might find second hand. Phil Askew, who runs the site, is well regarded as an impartial reviewer. The only thing he won't tell you is which camera to buy but he will point you in the right direction.
If this is your first encounter into the world of digital photography, best of luck but be warned it can be addictive.