Also, should I move the icons to a sidebar I'm not using, like "footer widget area one?"

I see that you already moved them to the sidebar before I replied. I think it's easier to move the widgets around with it set like that, but for someone else reading, it really depends on your theme and what you're trying to do. For example, if the sidebar isn't shown on some pages for a theme, then the widgets won't show up on those pages either.

I am trying to get the hyperlinked social media icons from the Primary Sidebar on the left up to the top right of the page.

To do that, first you need to view the page source and figure out the unique ID for the widget you'd like to move. I looked at http://nutalon.wordpress.com/ and I found that your social media icon widget has an ID of "text-2". Here is some example CSS that uses absolute positioning to move that widget to the top of the page:

#text-2 {
	position: absolute;
	top: 65px;
}

#text-2 .widgettitle {
	display: none;
}

Note that you can adjust the top value to move the widget around on the page and you can add values like width or margin and padding if those things are needed.

I'm new to CSS - I've only copied what I've seen in other threads. I've tried copy/pasting code from other threads, but it shows up as text, not to mention it doesn't work, am I missing something?

Welcome! Copying and pasting from another thread will only work if that thread was also about the Oxygen theme. It's good that you tried some things though!

I'm new to CSS - I've only copied what I've seen in other threads. I've tried copy/pasting code from other threads, but it shows up as text, not to mention it doesn't work, am I missing something?

Also, should I move the icons to a sidebar I'm not using, like "footer widget area one?"

http://www.nutalon.wordpress.com

Thanks everyone in advance, I know you're all geniuses.