How can I send all error messages and maybe some system initialization messages to the papertrail log? Browsed around quite a bit and checked out the papertrail exmaples but still not sure.
That depends on how these are produced. When they are created correctly they will end up at the log handler(s) automatically. If not, then you have to change how you produce these messages.
These are by default channeled to all instantiated log handlers.
To test your logging you may want to also instantiate a SerialLogHandler and see whether the desired info turns up there at least.
And for completeness, of course your Argon needs to be connected to the internet to for Papertrail to work and depending on your SYSTEM_MODE and SYSTEM_THREAD maybe even to the Particle cloud.