So my kids have been installing software & removing it and suddenly my mac is acting very weird.
Yeah, if this is a machine you depend on for your own use, don't let them do that. Give them their own separate accounts with no admin privileges. If it's a family machine and you want to allow them to play like this, you'll just need to be prepared for this sort of thing.
I want to wipe it clean and reinstall the OS.
Basically, you just need to back up your entire user folder. (Or, if there are multiple users, all user folders.) You won't need all that, but you'll have whatever you need. Also, make sure you've got a minimum of two backups before erasing the hard drive. Remember that, once you have erased the hard drive, a single backup becomes an only copy, which is highly dangerous.
You'll need to reconfigure iTunes after the reinstall to look for your library on the external drive, but other than that, it should just be a matter of importing the desired data from the backup. If it were me, I would import manually, and would not import any settings files that I could avoid importing, but you could also use the setup assistant after installation to import from the backup.
