This worked for me, and was the easiest and fastest way I could find to go from 1.1.1 to 1.1.3 Jailbroken.
First, make sure you have firmware 1.1.1 installed on your iPod Touch. If not, put your iPod Touch into recovery mode by holding down the Home key as it starts up. Click Restore while holding down the Options key, and select a copy of the 1.1.1 firmware. Once this is done, you can turn your computer off, because you don't need it for the rest of the process.
Turn on your iPod Touch, open Safari, and open up http://jailbreakme.com . Scroll to the bottom, and click Install AppSnapp. This will jailbreak your iPod. When it finishes, open up the Installer app. Tap the Sources tab. Add ijailbreak.com/repo.plist as a source by tapping Edit, then Add. Then tap Refresh. Tap Install, scroll down to System, and select BSD Subsystem. Install it. Now, go back to the Install tab, scroll to iJailBreakMobile, and select iJailBreakMobile 1.1.1. Install it, then exit Installer by pressing the Home button.
Now, run iJailBreak. Just let it sit and run... it will automatically do everything, and then restart when it has finished installing 1.1.3. If you get a whole bunch of error messages (file not found, access denied, etc.) and it stops without downloading the 1.1.3 firmware, reinstall BSD Subsystem and iJailBreakMobile 1.1.1. Restart your iPod Touch, and then try running iJailBreak again.
After it restarts, to finish up, open Installer, tap Install, scroll to iJailBreakMobile, and install Remove 1.1.3 Upgrade Data. This will delete a few temporary files used during the upgrade process.
You now have a Jailbroken 1.1.3 iPod touch!
**Optional: iPhone Apps
First, install the BSD Subsystem again. Then go to iJailBreakMobile, select iPod touch Software Upgrade. Exit installer, and run the iJailBreak application. When it is done, you will have Maps, Photos, Stocks, Weather, and Mail installed. Open up Installer again, and install Fix Mail Passwords from the iJailBreakMobile section.
First, make sure you have firmware 1.1.1 installed on your iPod Touch. If not, put your iPod Touch into recovery mode by holding down the Home key as it starts up. Click Restore while holding down the Options key, and select a copy of the 1.1.1 firmware. Once this is done, you can turn your computer off, because you don't need it for the rest of the process.
Turn on your iPod Touch, open Safari, and open up http://jailbreakme.com . Scroll to the bottom, and click Install AppSnapp. This will jailbreak your iPod. When it finishes, open up the Installer app. Tap the Sources tab. Add ijailbreak.com/repo.plist as a source by tapping Edit, then Add. Then tap Refresh. Tap Install, scroll down to System, and select BSD Subsystem. Install it. Now, go back to the Install tab, scroll to iJailBreakMobile, and select iJailBreakMobile 1.1.1. Install it, then exit Installer by pressing the Home button.
Now, run iJailBreak. Just let it sit and run... it will automatically do everything, and then restart when it has finished installing 1.1.3. If you get a whole bunch of error messages (file not found, access denied, etc.) and it stops without downloading the 1.1.3 firmware, reinstall BSD Subsystem and iJailBreakMobile 1.1.1. Restart your iPod Touch, and then try running iJailBreak again.
After it restarts, to finish up, open Installer, tap Install, scroll to iJailBreakMobile, and install Remove 1.1.3 Upgrade Data. This will delete a few temporary files used during the upgrade process.
You now have a Jailbroken 1.1.3 iPod touch!
**Optional: iPhone Apps
First, install the BSD Subsystem again. Then go to iJailBreakMobile, select iPod touch Software Upgrade. Exit installer, and run the iJailBreak application. When it is done, you will have Maps, Photos, Stocks, Weather, and Mail installed. Open up Installer again, and install Fix Mail Passwords from the iJailBreakMobile section.