What SVN command can I use to see the branch I'm currently using?

Try the following:

svn info

This will give you the URL your workspace is a checkout of, relative to where you are in the workspace. You should be able to see the branch in the URL.

With some regex, you can get just the branch name:

svn info | grep '^URL:' | egrep -o '(tags|branches)/[^/]+|trunk' | egrep -o '[^/]+$'

'svn info' from https://edoras.sdsu.edu/doc/svn-book-html-chunk/svn.ref.svn.c.info.html

The branch is what shows up in the URL below:

$ svn info foo.c
Path: foo.c
Name: foo.c
URL: http://svn.red-bean.com/repos/test/foo.c
Repository Root: http://svn.red-bean.com/repos/test
Repository UUID: 5e7d134a-54fb-0310-bd04-b611643e5c25
Revision: 4417
Node Kind: file
Schedule: normal
Last Changed Author: sally
Last Changed Rev: 20
Last Changed Date: 2003-01-13 16:43:13 -0600 (Mon, 13 Jan 2003)
Text Last Updated: 2003-01-16 21:18:16 -0600 (Thu, 16 Jan 2003)
Properties Last Updated: 2003-01-13 21:50:19 -0600 (Mon, 13 Jan 2003)
Checksum: d6aeb60b0662ccceb6bce4bac344cb66