Is this a desktop or laptop? If it's a laptop, you will most likely need to replace the HDD with the SSD unless you have a laptop with multiple 2.5" drive bays. If it's a desktop, then you can disconnect the drive (leaving it in your case) and connect the SSD.
If you've performed the upgrade on your HDD and it's activated correctly, you can download the media creation tool:
Windows 10 Media Creation Tool
Pick the one you need. Make either a bootable USB key or DVD. With this you can perform a clean install of Windows 10 on your SSD. It will ask for a Product Key twice during the install, just click Skip each time it asks. The installer will proceed and eventually you will land at your desktop. Once the computer connects to the internet, it should activate without issue, as long as your upgrade activated like I mentioned above. At this point you can proceed with installing any drivers you need, Windows Updates, and software you want. Once you have your system running to your satisfaction, you can shutdown, reconnect your HDD and restart if this is a desktop. All your user files should still be intact, you can move these to your SSD if you wish, or if you intend to keep them on the HDD, you can back them up to a backup drive. Once you have everything backed from the HDD, you can wipe the drive and set it up as data storage. If it's a laptop, you'll need to remove the SSD, and install the HDD once again and boot from it so you can back up your data.
You'll notice that with either contingency, you left your working install of Windows 10 on your HDD alone until after the successful clean install had completed. This way if something goes wrong, you have a fall back and all your data is still intact.