Clone / Copy Oracle 10g & 11g
Welcome to the world of Database Administration... and good luck. You're going to want to read up on as much Oracle documentation as you can, as well as other good technical sources (O'Reilly has always been good), and subscribe to lots of Oracle blogs. 

I'll answer your questions here, but you're really going to want to get a solid foundation in RMAN ( For 10gR2: http://www.oracle.com/pls/db102/portal.portal_db?selected=4 ).

1.Do you have to shutdown your Oracle DB when you want to make a copy/clone of it?

Yes and no. It depends on if your database is in archivelog mode. If the database is archiving its logs, the backup can be done while your database is online, though you may notice some performance degradation during the backup, so it is still a good idea to schedule the backup during a non-critical time of day. If the database is not archiving the logs, then you must shutdown the database cleanly in order to make a copy of it (any other way will result in a corrupt/incomplete restore).

2.A good beginner example

This is hard to do without knowing your environment. There are various RMAN commands that will happily clone a database on the same server, but when you get into moving a clone from one server to another, you have to go a different route. So without knowing your environment, I can't really tell you a good example.

That said, essentially your goal is to do the following: 
 - Get a good copy of all your datafiles
 - Get a good copy of all your archive logs
 - Get a good copy of your database parameters and control file
 - Create a new database with the settings from your old database (or reuse an existing database)
 - Copy over your datafiles and archive logs
 - Restore from your controlfile, and recover until there are no archives to process.

RMAN does a lot of the work for you with some fairly simple commands, so I suggest learning how to use it effectively, but you can always do the hard work without it. (For a long time at  a previous workplace, we did this with some shell scripts and such. Fun it was not, and was easy to screw up.)

3.Does a GUI exist for cloning an Oracle Database?

I think the thing that comes closest would be Oracle Enterprise Manager (GRID). It offers several automated backup/restore options as well as cloning features. Keep in mind, however, that it uses RMAN under-the-hood, and you should never use a tool blindly without understanding what's going on underneath. Sooner or later, you'll need to delve down into the command line because the GUI won't do what you want it to do, and without a good understanding, you'll be stuck.