Stay inside security and walk. There is a tram outside security, but going through security again would cost you any time savings.

If you go to http://www.flychicago.com/ohare/terminals/terminal.shtm, you can see the connections between the terminals. They are all airside. From American, at first follow signs toward concourse G. When you get to the base of G, keep going and you'll go through a hallway into Terminal 2. You'll pass the entrace to concourses E and F and then eventually get to a hallway going to Terminal 1. Pay attention at the end of this hallway, you need to turn left down a narrow hallway to get to the airside part of Terminal 1. If you go straight, you'll be exiting security.