On the off chance you become consistently epically successful you should be prepared to move to dedicated hosting and to split the application between servers if necessary. This will take extra development time especially if you don't really know what you are doing. But, I think it is a worthwhile skill set for an ambitious young web entrepreneur to learn.
I think it would be optimal to immediately try to monetize traffic to determine whether you really need to pursue outside funding. I estimate that you can recklessly fund the adventure on even a lower limit credit card (or just get an Amex charge card if you're feeling particularly reckless).
As everyone says it is a good problem to have. But, as a matter of full disclosure, I should note that I don't really know much about this. I am an as yet unsuccessful entrepreneur. Good luck.
1) Which sites have seen their traffic grow to millions of visitors daily within a few days of launch. I can't think of any, but you must have a few in mind.
2) What did they do or fail to do to handle that growth?