There is no way around multi browser testing. Emulators, some websites and the like can help, but ultimately, you must test on the real thing, by installing different browsers, different versions and on different OSs and looking at the results, adjusting any possible difference.
A couple of browsers side by side is a good starting point. The portable ones are very good at that (you can have multiple, even different versions at the same time). I would suggest at least having Firefox, Chrome and IE as the bare minimum (IE being the only complicated, just because you can't install it side-by-side), as well as any other browser your potential users might use.
Also remember to test on other OSs too. Windows Mac, Linux, Android and iOS are the most common, and those are what you should test ideally.