I currently use OneDrive across all of my devices and computers. With HP Connected Drive ending its service, will Onedrive continue its service? Should I back up everything from Onedrive to my computer?
OneDrive isn't going away so you should be fine, here is what the support document says and the link if you would like to see all of the FAQ's:
"With the launch of Windows 10, HP is no longer offering the HP Connected Drive service. Our recommendation is to use Microsoft OneDrive as a central file service across all your devices. HP strongly encourages you to download any files in your Collection that you want to keep before the Connected Drive service is discontinued."