Another option is to use the LogHandler. It handles both system logs and can handle user logs as well. You just need to replace your Serial.print with Log.info calls. It's also better because it's thread-safe, and does things like timestamp the entries for you.
I send the logging for all of my devices over an encrypted TCP connection to my server which stores them in a MySQL database.
Another option is to use the Papertrail log handler to log to the Papertrail service: