Soon I'm going to wipe my computer, and once I'm done I'd like to have Windows 10 installed. I know Windows 7 users can upgrade to 10 for free. Since I'll have to reinstall Windows anyway, is there a way to take advantage of the free upgrade to Windows 10 without installing Windows 7 first?
Windows 10 licenses are linked to the hardware (the motherboard). To obtain a valid Windows 10 license, upgrade to Windows 10 before wiping your old Windows 7 once. Windows 10 will activate when it finds your valid and correctly activated Windows 7 license. It will then store your hardware ID on a Microsoft license server. After that, you can re-install Windows 10 on that machine from scratch and it should activate (when connecting to the Microsoft license server via internet).
Note that you are not allowed to use this Windows 7 license on a different machine as long as you use Windows 10. Also to my knowledge you cannot use the same Windows 7 license to activate Windows 10 on a second machine. Therefore this procedure will effectively link your Windows 7 license to this specific hardware. For a different motherboard, you will need a new license.
Edit: I just read that with Windows 10 Version 1511, you have more options. You can install Windows 10 in parallel to your old Windows 7. To do so, use the Media Creation Tool for Windows 10 Version 1511 to download a Windows 10 image and either copy to a USB stick or burn to a DVD. When executing the Media Creation Tool, choose to create the media for a different PC. This allows you to choose language, edition and architecture. There are a number of Media Creation Tools out there, be sure to use the correct one: Windows 10 Version 1511.
When booting from the installation medium (press a key early in the boot process to really boot from the installation medium and not the installed OS), you can enter a Windows 7 or Windows 8 or Windows 8.1 key and start the installation.
Since you want to change hardware: I don't know if this type of installation requires a working Windows 7 copy. In your case it seems worth trying if it really does before re-installing Windows 7 onto your new hardware only to do the update. You could also try keeping the old, probably defunct installation of Windows 7 on one partition and install Windows 10 to a new partition (or a new hard drive).