The fact that you gave them remote access to your computer is bad. They may have left some little nasties around, like a backdoor to get back in to your system in the future or a keylogger to steal your passwords. They may also have accessed any of the unencrypted data on your hard drive.
To start, erase the hard drive and reinstall everything from scratch:
How to reinstall Mac OS X from scratch
Alternately, if you use Time Machine (or something similar) for backups, you can restore your whole system to a point in time prior to giving them access.
Once you have a clean system, you need to start doing things to protect yourself. Change all passwords for all online accounts. If you have any sensitive financial data in unencrypted form on your computer, such as credit card or bank account numbers, notify the company/bank of the potential breach in each case.
These people may also have enough information at this point to effectively steal your identity, so it wouldn't be a bad idea to sign up for a credit monitoring service, like LifeLock. (I don't actually know if LifeLock is any good or not, and don't recommend for or against them; that's just the one that most people recognize, so I used it as an example.)
DO NOT send the check!
(Fair disclosure: I may receive compensation from links to my sites, TheSafeMac.com and AdwareMedic.com.)