I want to thank Sean Beck for bringing to my attention that I did not mention what port to use in the Arduino IDE in my first post. I thought this should be easy enough to remedy, but after looking in the Arduino IDE there was no port assigned. In the status area, device /dev/ttyUSB0 was being used but when I checked my /dev directory that device wasn't there.
So what port is the Trinket on? I went to the Adafruit forums and found the answer which turns out to be no port.
From Adafruit Forums: Forum Index > Supported Products & Projects > USBtinyISP Trinket Help
The Trinket does not emulate a COM port. Use the Trinket flavored version of the IDE and select Trinket as the Board type. The IDE will look for a USBtinyISP instead of a COM port for programming.
This forum answer talks about the Trinket flavored version of the IDE which is now outdated since the newer IDEs can be made to see the Trinkets.
For the Arduino IDE, just make sure the Programmer option in the Tools section is set to USBtinyISP. Also do not forget to press the reset button at the end of the Trinket before uploading the sketch.
To learn more about the Trinket, visit Adafruit Trinket Tutorial.