This is easier now than ever. Lately I've had a lot of luck writing something once and having it work with IE10, Chrome, and Firefox immediately. That leaves on

This is easier now than ever. Lately I've had a lot of luck writing something once and having it work with IE10, Chrome, and Firefox immediately. That leaves only fixes for IE8 and IE9. By designing things to be responsive from the get go, mobile is 98% supported off the bat too with only small fixes needed thanks to Webkit's majority. I would suggest starting with a base such as Bootstrap (or another framework of your liking) and getting a Browserstack account. Testing becomes much easier with Browserstack and they were offering three months free with a coupon, so give them a try.