Q: Apple ID hacked

If someone hacks into your Apple ID, what can they do with it? What should I be watching for?

I tried to log a security incident, but it seems I cannot... * there isn't even a support topic OR community topic about it!!! (Hence using this category...)

History:

I received 2 emails yesterday saying someone was trying to log in, but that 'it may have been an email entered by mistake. Woke up this morning to another one saying the reset was requested again, and then one confirming the change had been made.

I had to reset again myself, etc. 

I called and was told 'Apple takes security very seriously' yet they don't log and incident or do any type of investigation. No wonder Apple has such a terrible reputation for security! (When I mentioned this to the agent they were 'sorry that I was having this problem' (?) and wheni mentioned the rep and that there are many news stories relating to security issues/breaches, she simply said 'oh, I am unaware of news.'