I remember when I got my first pair of contacts, I was in 10th grade. My mom took me to Sears optical in the mall, and they wouldn't let me leave without making sure I could put them in. I had to sit at a little table at the edge of the main mall thoroughfare and try to stick my finger in my eye, it must have taken me 30 minutes, with what felt like everybody watching me. It was unpleasant. But now I don't even use a mirror. I pull my bottom eyelid down with one finger while moving my eye all the way up, and put the contact below the iris, then let go of my bottom eyelid and look back down. Works every time.
Oh, make sure the contact isn't inside out, if it is you can feel it.