Once you install OneDrive on the old lappie you will have a OneDrive folder appear in windows explorer. Just MOVE all of the files from their current location on the lappie to that folder are you are done. They are both ‘in the cloud’ and on the local hard drive and OneDrive does the syncing. As always the way to do it is to try it with a single file first and once you are comfy with it try moving more
There is a much newer version of OneDrive that lets you pick which folder you want to sync but I’ve not tried it yet and it sounds like it might be a bit chaotic. I like simple. OneDrive folder = synced
edit: and of course, you need to be using the same microsoft account to log into both lappies