The approach I use is to write the CSS for my site and test it only in Google Chrome. You can use Firefox 4 too. The reason I use these browsers is that if my CSS looks good in them, I'm pretty sure it's standards compliant and has a great chance of looking good as newer browsers come out.
Then, I take a look at my site in IE. Now, thankfully, you don't really have to worry about IE6 any more. Almost everyone's ignoring it. So you have to think about your audience. Do you think most of them would use IE9 or IE8? If you're ok with IE9, then it's almost certain your site looks great without any modifications because it's a very standards compliant browser. If you need IE8 compatibility, then add another CSS stylesheet and make it so it only shows up if the browser is IE. Your tag would look like this:
<head>
<link rel="stylesheet" type="text/css" href="your-main-style.css" />
<!--[if lte IE 8]>
<link rel="stylesheet" type="text/css" href="ie8-and-down.css" />
<![endif]-->
</head>
So now you can put any style fixes in that stylesheet without hurting the style in your main stylesheet. Keep this to the absolute minimum so that it's easy to maintain.
As an aside, if it's just a site for you to have fun with, pick a browser you like and make sure it works in that. Then just put a message at the bottom saying what browser displays your site best.