No. While that method would copy over all your files, it would not copy over the MBR and other hidden files needed to boot an OS.
No, what you want to do is called "cloning". You use the cloning software to clone your old drive to another drive. Then remove the old drive or set the new drive as the new boot drive. And voila! Your system just the way it was before the stability issues.
Luckily for you, such cloning software is free:
http://support.wdc.com/product/downloaddetail.asp
I downloaded that tool, and it starts just fine, thank you. But will it guide me through everything? Will I have any data loss? Would I need a DVD drive plugged in? I only have two sata wires currently, so I would unplug the DVD drive one, and connect the new HDD to it for transfer.