You should plan on checking out the Android - Processing Wiki regularly. For now you can just follow the instructions for downloading Processing and the Android SDK. I've done this both on a Mac running OS 10.7 and on Windows 7. Having done this on both I can say it works much better on my Mac. There is a buggy bit about running it on Windows, but I've already figured it out, so you won't have to.

Just follow the instructions for downloading and installing the Android SDK on the Android - Processing Wiki. Also know that you may be prompted to download and install the JAVA JDK. It's not a big deal, just expect to do that before finishing the install. Processing itself does not have to be installed. You just unzip the download and run it from the included folder. If you only want to use processing and don't care about Android this is really all you need to do. You can even begin using Processing with your students while you wait for your tech department to approve the install of the Android SDK. You just won't be able to preview your work in the Android Emulator.

Mac Issues:

I haven't had any yet. If you do let me know and I'll include them here for future users.

Windows Issues:

Problem - Emulator Never Launches at all: 
You just installed everything didn't you. Restart your computer and you should be good to go.

Problem - Emulator Starts but your Project doesn't get there
This one is more problematic. Follow the steps below and/or watch the video walkthrough.
  1. First time - Try to launch a sketch in the emulator. Wait for it to start all the way. The close the Emulator Window.
  2. In Processing goto Tools-Check exactly the menu structure and insert here
  3. In AVD check menus and insert here
  4. Start the Emulator and wait for it to launch all the way.
  5. Go back to Processing and "Run in Emulator". A second emulator window will open up, but your sketch will launch in your first window.
  6. After the second emulator is up and running all subsequent sketches will launch there. (and you can probably close the first one)

Windows Workaround