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Adobe Browser Lab
The best tool
The best way is to use some tool to test here is the link which can give you better idea.
http://www.1stwebdesigner.com/design/tools-browser-compatibility-check/
- Ameen.S
I would recommend you start checking your user base (what browsers they use) and then define the list of browsers you want to support.
The rest really depends on your approach, usually I start testing on firefox/chrome (99% of the time, what works on these2 browsers will work on all modern browsers); once all bugs are fixed, you can switch to IE, you might want to check the common problems in IE, here is a very good post with the list of common bugs and how to fix them: http://net.tutsplus.com/tutorials/html-css-techniques/9-most-common-ie-bugs-and-how-to-fix-them/