Switching Primary HDD, Will this work?

If the new drive is an IDE (40-pin ribbon cable), drive, you can use free cloning software, copy your drive, and boot away.

If it's an SATA drive (7-pin cable), there is some work to be done. What the work is depends on what operating system you are running. Can you tell us what motherboard you have, or make and model and what kinds of disk controller ports it has? EDIT : just noticed hardware specs in your sig. But I don't see the motherboard. And with all that tasty stuff, why on Earth were you using an old IDE disk?

Best shot for changing drive types would be to backup to an external drive using something that has the Restore to Dissimilar Hardware option, like EASEUS ToDo backup. Most of us here would do a full install on the new drive, so it's sparkly clean, instead of a clone, though.

Do you have the OS disk? Is it a real Windows distribution disk, or a manufacturer's recovery disk?

I'm running Windows 8 Pro 32-bit, and I used the Windows 8 Upgrade assistant, and don't have a physical disc.
It is indeed a SATA HDD.
My motherboard is the ASROCK N68C-GS FX
The mobo has 1 IDE port and 4 SATA ports. I have an IDE HDD and Optical Drive, with a Two IDE Ribbon cable connecting them.
Is it somehow possible to just copy and paste everything from C: to my new drive and then change the boot order? Sounds too simple to be true. Thanks for your help!
~Guy  

Copy and paste; no.

You will have to "clone" the drive. If you don't have a physical distribution disk, then you can't do a straight clone followed by a repair install.

So download something like EASEUS ToDo backup and do an image backup to your external drive (you have a backup drive, yes?). Then remove the IDE drive and set it aside for emergencies, install the new SATA drive, and boot the backup utility from the bootable CD that you made. Now restore from the external drive to the new drive, checking the "restore to dissimilar hardware" option. If the utility is nice to you, it will replace the low-level IDE drivers with SATA drivers and you will be able to boot.

You may have a licensing issue. Where did that old drive come from? If it's from a machine that shipped with the OS installed, and you put it in the new one so that you could do an upgrade install to Win8, it may not be licensed.