The first time I remember seeing shooting stars was at a Girl Scout camp in Michigan. As the articles I linked to above point out, a rural area without a lot of artificial light is the best place to see the night sky. Campsites are great for that, and so are the small towns in Ohio where my brother and I went to college! The article from the Tribune proposes taking a road trip to a good site, and provides links to some dark sky maps to help you find the best locations. The closest one I found was in New Lenox, although you could easily go as far as Wisconsin.
My most organized experience with watching a meteor shower was during an astronomy class in college. Our professor determined that the busiest part of the shower would be in the early hours of the morning, between midnight and dawn. We bundled up, brought blankets and sleeping bags and interested friends, and settled down in an open field to watch the sky. There was a bit of a wait before we saw anything, so it was good to be prepared for lying on the cold ground.
We were chatting and joking at first as we waited for the meteors. When someone saw the first long, bright streaks of light we all started pointing and shouting as if we were watching fireworks. I remember that as the meteor shower really peaked, and we saw one meteor after another streak across the sky, everybody quieted down to just watch and enjoy.