If the user is complaining the hang problem ..then the experience of a dba reflects the work style that he is going to perform and basically as the rule suggest first try to connect to the database itself and fire some query to check whethere you are allowed to connect or not if you are connected then check for any locked_objects by checking v$locked_object and dba_waiters and dba_blockers.then you have to eliminate the things which are working properly and targetting yourself to the weaker place and then check at os level that which process is consuming the most of the time and then analyze the problem if the problem relates to a single user process then check what that user is doing by checking the sql statement he is firing.than so on

If problem is pertaining to database & a user dont have acess to database level . One can check the process ( taking long time ) through the enterprise manager.