Oracle 11g SYSDBA login
It depends on the GUI but they all should have a way of selecting the SYSDBA privilege. Toad has it appear as a drop-down box called 'Connect as:' as does Oracle Enterprise Manager. For SQL Developer you use the 'Role' box when creating a new connection.

As far as / as sysdba is concerned it just means that operating system authentication is being used so no password is needed. You can only use it if you are logged in as the oracle user or a member of the dba group (or for Windows, I believe as an administrator).

When you grant a user the SYSDBA privilege it only means they have the option to log in with the SYSDBA privilege and not that they will have dba privileges when they log in normally. The same is true for the SYS user: you can log in as SYS without the SYSDBA privilege and so lack the ability to shutdown the database, for example.