This will be easy, you don't need an external drive to transfer stuff over, but it would be a good idea to get a portable external USB powered drive to use with TimeMachine after you get the mini up and running.

Start with the mini off.

Boot the old computer into Target Disk Mode, it's considered the Target.

Connect a Firewire 800 cable from the G5 to the mini, it doesn't come with the computer, you may need to get one.

If you are using the same keyboard and mouse for the mini, you can disconnect them from the G5 before booting the mini, and after booting the G5 to TDM. You can disconnect the monitor also after going TDM and connect it to the mini before powering up the mini.

After getting the keyboard, mouse, and monitor hooked up, boot the mini.

If it's a new mini, once you start the mini, there will be an option to transfer data from another mac.

If it already has been started at some point up and has a user, you can use Migration Assistant in the Utilities folder to transfer users and data from the G5.

When you connect the G5 in Target Disk Mode, the hard drives will show up on the mini like external drives.

What does the connector on the end of the monitor cable look like. If it's like on the left you will need an adapter, if it's like on the right there is already an HDMI to DVI adapter in the box.