1. Set out your personal budget and choose if you want to buy new or used.
2. Make a list of "must have" features and compare possible cameras against this.
e.g. DSLR or compact. Fixed lens or zoom. Fully auto or manual controls. Viewfinder or just LCD screen. Sensor size. Compatibility with other equipment owned.
3. Make a list of "desirable" features and compare.
e.g. large zoom range. Macro mode. Good low light capability. Wide accessory range. Inbuilt special effects.
4. View reputable websites (dpreview, trustedreviews, cnet, etc) for reviews and look at video reviews. Look at users reviews, e.g. Amazon. This will provide you with potential problems or issues as well as opinions on image quality, autofocus, etc. For example, I was looking at a Canon compact only to find out that several users mentioned shutter problems.
5. Create a shortlist of cameras that you like that are within your budget.
6. If you can, go to a camera shop to examine and handle shortlisted cameras to see if they meet your expectations.
7. Compare the market for your chosen camera. Look at any warranties carefully. If buying used on ebay, don't rush to buy, instead create a search and keep looking at the market. Check out "buy now" options on ebay against camera shop offers and Amazon.
8. Buy your camera and check it out immediately for any issues or problems within the returns period.