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Most important thing to back up is your personal data which you can do via drag and drop ie music,pictures, documents etc
If you wish to keep you installation ready to copy to a new drive you will need to take a system image of your hard drive ready to transfer over to a new hdd but be aware that as you are having problems and suffering corruption of system files you will also transport that corruption to a new hdd so may not work as expected so may be time for a clean install
Heres some links that may help which ever way you decide to go
Clean Reinstall - Factory OEM Windows 7
Backup Complete Computer - Create an Image Backup
Imaging with free Macrium