Export High Resolution Images

After spending a week, night and day, learning illustrator to create a logo for a start-up business, I am unable to find a way to export high resolution images/graphics. I've searched for solutions online, including this forum, and tried all file types, settings, etc. yet nothing has worked.

I bought illustrator assuming it is the preeminant graphics program.  Have a really wasted a week of time with this program? Is there no solution?