The first thing you must do is make sure your system is up to date:
sudo apt-get update
sudo apt-get dist-upgrade
Install any packages that are out of date. After that, time to get the update manager:
sudo apt-get install update-manager-core
Since we're upgrading from Intrepid Ibex to Jaunty Jackalope there shouldn't be a reason to modify any other files like you might when upgraded from the LTS. So get straight to the matter by running this command:
Follow the on-screen instructions. If you're not an advanced user, I'll recommend just taking whatever defaults you are prompted with. Once you're done, reboot!
A couple of things went wrong with my upgrade. First, my Passenger Phusion install went awry. This was 100% my fault though for not upgrading the right way. Just in case you missed it, follow the instructions on the official blog.
The second thing I needed to fix was my rmagick install. A key library got removed and so it was back to the drawing board. Luckily, I found some easy instructions courtesy of Rob Britton.
sudo apt-get install libmagickwand-dev
sudo gem uninstall rmagick
sudo gem install rmagick --no-ri --no-rdoc
And with that you should be done, and hopefully enjoying your upgraded stack. It also couldn't hurt to do a sudo gem update just in case you have some old crud hanging around.