You can use a website like browsershots, Browsera, Cross Browser Testing, Multi-Browser Viewer, BrowserSeal, and Multi Browser vmWare Appliance
You shouldn't need to be checking that many browsers, however. With the exception of Internet Explorer which seems to make radical changes with every release, most of the modern browser only make incremental changes that rarely break their consistency in rendering pages. This means if you test in the latest Firefox, Chrome, or Safari it almost certainly will work in an older version. Additionally, most of the users of these browsers update their browser to a newer version quite rapidly compared to IE users who tend to hang on to older versions for far too long.
To keep your testing simple I would do the following:
In Windows test Chrome, Firefox, IE7-9. You can test the older versions of IE using either a tool like IETester or use the Developer Tools in IE to simulate older browsers.
In a Mac test Safari, Firefox, and Chrome.