I learned about them by learning Perl, and using it to create a configuration file parser. If you wanted to do something similar these days I would suggest Ruby, which has a similar hook for RE in it. Once I learned the basics, I found many of the tools I'd been using had great RE support, if you knew to use it. It is really very powerful.
I highly recommend the Regular Expressions Pocket Reference that others have mentioned. It will be much simpler to keep around to look up something quickly, as you will need to do. Not to mention that many applications have slightly different ways of implementing some parts of it (ie, whether you should have to backslash escape parenthesis).
If you do much text processing, I recommend learning RE. You will appreciate it.
