I tried tracert and traceroute, then a lot of garbled versions of anything I could think of. Is there a command for this built in or do I have to install something?

In most *nix systems, it's in /usr/sbin. I couldn't find one anywhere in the Sharp ROM, but I suspect that you can find one compiled for ARM Debian, Familiar or Netwinder that will work on the Zaurus.

Of course, there's always this app:
http://killefiz.de/zaurus/showdetail.php?app=728

I got mtr to work after I went into SU. Thanks!