You have to go into the code.
InDesign does a reasonable job of converting your text into HTML/CSS, and if you carefully prepare your document it actually does a *great* job--that is, clean and ready to be tinkered with manually. Any way, weird stuff to appear in the epub needs to be taken care of that way.
There is no way around, to be able to produce a reasonable epub using ID *requires* you to know more of the internals of epubs.
Off-hand I don't know what small caps get translated to into CSS. It ought to be possible, and it's possible that small caps DO get exported correctly but the software you are viewing the epub with doesn't recognize it.
but please don't tell me to go into the code, because I have no idea how to do that.
Then I strongly suggest you learn. It's not that difficult. Additionally, you really need to know how to use styles in InDesign as opposed to just styling things manually.
Anne-Marie Concepcion has some excellent videos on Lynda.com. This link will get you a one week trial: http://bit.ly/fcGpiI
BTW, I'd love to be able to help you but without seeing the markup it's pretty difficult to know what happened and it would certainly help to know what version of ID you're using and how you're viewing the EPUB.
Off-hand I don't know what small caps get translated to into CSS.
I just did a tiny test and InDesign did use this in the CSS rule for my small caps character style on export, but it had no effect in Digital Editions. It worked correctly when I opened the HTML in Safari, and also when I used Kindle Previewer to simulate Kindle Fire and the iPad Kindle app. Didn't work on the plain Kindle, though. So it looks as if it is the ePub reading software that is failing here, not InDesign.
But I saw no sign of any small caps turning into italics in any of these readers.