Help with start up. new iMac 27" with Marericks installed has been giving me trouble with IDs and passwords. More than one account downloaded from older Laptop. Now the computer has the turning wheel in from of gray screen and won't go any further when

Question

I recently started using my desktop iMac 27" and have been having problems since installing Mavericks, not saying that Mavericks is the problem per se.  I think I have too many accounts with too many names, with too many Apple IDs and too many passwords, email passwords and Apple ID passwords, and keychain passwords, etc.  Can't keep them straight even though I write everything down.  The screen has been making me log in and log out with Account name and passwords after everything I've been doing, also confused about Users and Groups and how to sync everything. I finally just "shut down" everything and the screen was black.  When I turned the computer on again, the circular gear in the middle just kept turning but nothing happened after that.  Then I held the on button in the back until the screen went black again. 
What should I do?  Is there a way to combine all my accounts into one account with one Apple ID?  I should say that I also have trying to use iCloud and it has different IDs and passwords. Also when I try to log in with one account name, I can't enter anything I write,and when I type on the keyboard, nothing appears on the screen, although I can receive emails, I can't send them. 
In another account, what I type does appear on the screen, but I think it's an earlier account and doesn't translate aver to the other account. ALTHOUGH i set up iCloud accounts, they don't appear any more.
Very confused.  Any Ideas about how I can get the computer to start up again and show a new screen?
Thanks to anyone who has workable suggestions.

Answer 1

If you really believe that your system has been compromised, here's what you do:
Disconnect your Mac from your cable modem;
Back up any documents on your system that are important to you;
Boot your Mac from the system installation disks that came with it (insert the disk, restart your Mac, and hold down the "c" key until you get the "spinning gear" icon);
Choose a language and click the arrow button to continue;
From the Utilities menu, choose Disk Utility;
In Disk Utility, select your computer's hard drive;
Click the "Erase" tab;
Click the "Security Options" button and select to have it overwrite all the data on the hard drive;
Click the "Erase" button and allow it to process;
Once the "erase process has completed (it will take a while), reinstall Mac OS X.
Or, if this is too much for you to accomplish on your own, take your system to an Apple Store and have them help you perform these steps. If your system was indeed compromised, this will remove any such hack. You can then set up a new user account for the computer, reinstall your applications (reinstall only from original disks or downloads from the company making the software) and documents, and reconnect to the Internet.
Note that when you reconnect to the cable modem, you may still get an IP address starting with 198. This is normal with some cable modems and probably not a cause for concern. It will not be an indication that your system is still compromised; that will not be possible if you perform all the above steps.
Regards.