You need to test your css in all major browsers, including IE (6, 7 and 8), Firefox, Chrome and even Safari and Opera.
You may be able to simply use the latest versions of Firefox, Chrome, etc., since they automatically update, and therefore users are less likely to use the older version (although it is not unknown).
IE is a problem because many users, especially corporate users, work with older versions.
Choose one, it doesn’t really matter to what extent it has good development tool support in order to develop and get a site that is good. Then check in others. What order you do and how much effort you want to spend depends entirely on your target audience. In my experience, Firefox with Firebug and even (I dare say) IE8 with its built-in developer tools is a good choice. The advantage here is that you reach most users with these. However, as the browser market becomes more fragmented, you will need to test others.
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