HELP in Sydney!

[Sose]

Folks, there is a chance I'll be flying to Sydney for work. If it is approved, I'll be there from the 13th until the 18th for work. However, since Australia is not a place I'd usually visit for work reasons, I'd like to stay an additional week.

This, of course, means I'll be paying for this week on my dime. So here's what I'd like to know from any members that are able to assist:
What affordable hotel could I stay close to Sydney or in Sydney (read, less $100 or less)? If this is not feasible, please provide me with suggestions. Keep in mind I know nothing about Sydney - which leads me to my second question...

What activities can I not miss? What places should I not miss? What things should I not do?

Any assistance/ideas/comments are welcome.

Thanks!