Easiest for a "quick check" is to open an ssh connection and "tail" the log into a console. Of course, as soon as you get any of this rigged up, it will run perfectly for months 
The logs are not persisted by default as /tmp/ /var/ is a RAM-based file system. Continuous writes to flash shorten its life very quickly.You can add a USB "stick" pretty easily. As I recall, the block-mount package, plus the appropriate USB drivers are required. If you've got a Linux-based box, you can format the drive there. FAT is a possibility too, but doesn't support things like permissions and owners.
has a link to the instructions, as well as answering a couple common questions around mounting file systems.
I believe the "stock" logger can log to a file with appropriate configuration. I personally use syslog-ng and logrotate. syslog-ng can enable remote logging over TLS.