Please note that if you have used anySIM 1.0 or other older free SIM unlocking solutions (anything except for anySIM 1.1. and iPhoneSimFree) to SIM unlock your phone, and you upgrade to 1.1.1, it will wipe out the SIM unlock and render the phone and text message part of your phone inoperable. Everything else will work though.
Check the firmware version that's reported in the status line at the bottom of the iNdependence window. If it doesn't say 1.0.2 (or lower) then you'll need to downgrade your firmware using the instructions above.
Activate your phone using the activation instructions above. Note that you should leave your phone jailbroken after activation.
Install SSH on your phone using the instructions below.
Ensure that your phone is connected to a Wi-Fi network (read the iPhone manual for instructions on how to set up Wi-Fi). Also ensure that it's IP address does not conflict with any other devices on your Wi-Fi network (sorry, I can't offer network admin help).
Get the IP address for your phone by launching the Settings application on the phone, navigating to Wi-Fi and pressing the blue arrow beside the name of the Wi-Fi network that your phone is connected to.
Select the Firmware tab and press the Pre 1.1.1 Upgrade button.
Enter the IP address of the phone and the root password (see below for the default root password if you didn't change it when you installed SSH).
Follow the instructions given in the dialog windows. If iNdependence gives you a warning about host verification failure and asks to edit your known_hosts file, choose the Yes option (which is now the default).
Once you are told to upgrade using iTunes, quit iNdependence, and select the Update button in the iTunes window (not the Restore button).
Wait for iTunes to finish.
Quit iTunes and relaunch iNdependence. In the status bar at the bottom of the iNdependence window you should see Firmware 1.1.1.
Congratulations! You are now ready to activate/jailbreak your phone using the 1.1.1 instructions.