How can I copy files onto an SD card with a Honeycomb tablet?
Getting to your external in File Manager HD is...not very obvious. The only way I'm aware of to get to it is to hit the Home button at the top right on the action bar (not the Android OS button in the bottom left) and then hit Up. This should put you at /mnt, so you can get to the external at /mnt/external1. The "Storage" shortcut on the left pane also takes me there, but I don't think it's always operated like that in the past. I might be mis-remembering there, though. If "Storage" takes you to /mnt/sdcard then you can hit "Up" from there to get back to /mnt.

As for copying, the external card was originally read-only by default. Now it seems to exist in some kind of weird limbo (at least for me) and I'm not entirely sure how the permissions are designed. File Manager HD shows its permissions as dr- (that is, read-only directory) but it lets me paste files there anyway. Nonetheless, if you're rooted you can use something like MicroSD Mounter as an easy way to remount it RW (it works fine for me).

I should probably also note that the above is on Android 3.2.2, after getting Verizon's LTE upgrade. The permissions may still be read-only for Xooms in other regions, but I'm not sure. Mine could also just be read/write because I'm using a custom kernel and boot image (don't remember if it was in there).