If you try to boot with only the old HDD installed, it is normal for that to fail. The version of Windows installed on that old drive is customized for the old machine it came from, and does not have all the device drivers needed to get the new machine going.
But if you try to boot the new machine with BOTH HDD's connected, the same thing happens. THAT is an important clue. What it means is that your BIOS is set to boot from the OLD HDD when they are both connected, and that's why it fails. You need to fix that by changing a setting in BIOS Setup.
To do this you should have both HDD's connected in your machine. To enter BIOS Setup, normally what you do is to turn on the system and immediately hold down the "Del" key while things pass by on your screen, until the opening screen of BIOS Setup appears. When you do this, WATCH the messages on the screen, especially at the bottom. You should see a message there about what key to push (hold down) to Enter Setup. On some machines it is not the "Del" key, and this is where you'll see what you really should use. So if it did not work, try again using the right key.
Once in BOOS Setup you need to find the page where you set the Boot Priority Sequence. For most people, you should set it to try your optical drive unit first, then the HDD that has your proper OS on it. In your case, OP, make sure it is set to use your NEW HDD as the second device choice. Then make sure there is NO other device in the list, especially NOT the old HDD. Now use the screen prompts to SAVE and EXIT, and your machine should reboot properly.
When it does boot cleanly, you should be able to see your old HDD as a drive in My Computer and access all its files. Although it has an old Windows on it, that does not matter at all if you have your Boot Priority Sequence set properly as above. So, you can use that drive with no changes to it if you want.
IF you want to empty it completely and start with a blank old HDD, you can use Windows' Disk Management to go to that old HDD and Delete any and all Partitions on it, then Create a New Simple Volume. This will create a new Partition of the size you specify and Format it so the HDD is empty and ready to use.
Personally, when re-using an old HDD that contains nothing I want to save, I prefer to use a utility to Zero Fill it. Boot and Nuke does a very similar process. These tools completely wipe old data from the unit and do something that helps the HDD to find and replace any questionable Sectors before it is re-used, which is a bit better than just re-Partitioning it with Disk Management. But after wiping the old unit this way, you will need to use Disk Management to Create a New Simple Volume.