On PC am I only able to move my game captures to OneDrive or can I move them to something...

If your game captures are already on your PC, then you can transfer them to whatever service you want to use. You don't have to use Onedrive.

I am having trouble with moving to OneDrive to be able to save my photos. It won't let me move the ones that say "On the Xbox network". when I try and start to move it, it doesn't get one in and says "Failed to upload one of the items. Sorry, something went wrong when I attempted to upload one of the items. The items still selected have not been uploaded."

I don't have an external storage device that I am able to move them to. I can only try and use OneDrive. When I go to OneDrive it doesn't show any of my clips or screenshots in any of the categories. They are connected under the same email and everything.

Well I believe it says 'copy to OneDrive' or 'share to OneDrive' or something on the Xbox captures menu, doesn't it? Can't remember exactly, but as far as I'm concerned it would just upload them all to my OneDrive, then I'd be able to access them on my PC/Phone.

The only way I know of actually moving a capture is to send it to your OneDrive. Press the menu button with the capture highlighted and select "share" then select "OneDrive". And then delete the capture if you don't want it to take up space in your Game
DVR.

I don't believe there is any other way to store your captures on your Xbox One without them taking up space on the cloud, as that is how they are uploaded. I hope I'm wrong though!

You can't, it's not possible. There is a new feature coming soon, that will let you upload your captures to Onedrive automatically, then you can move them to your PC, and do what you want with them. The option you have at the moment, is what you are currently doing.