Some error checking is always useful. Imagine if for some stupid reason your database structure changes and an insert fails. You need to catch this. I never output SQL errors to general users (admin level users ok) as it can give them help to perform SQL injection attacks.
As well as checking the SQL error I also use mysql_affected_rows. This will show how many rows have been affected/inserted which is useful for updates and inserts
I'd suggest that you use the return value of mysql_query instead of using mysql_error in order to determine that an error has occurred... From there i agree with CoderGoblin that mysql_affected_rows comes in handy...