Thanx for getting back to me. You say, "use the Sharing System Preferences to share out the folders," I don't know what that means. Please explain.
Sorry, I can't post any screenshots to make it easier, but I can describe it.
Open Sharing System Preferences (Apple Menu)
Enable File Sharing
There will be a pane at the bottom where you can add folders. Click the + button to add your "share"
On the "Users" section, add the user you will log into that Mac as.
The permissions will show what is set for that folder. Add your user to the "users" and give that user read or read/write. Unless you need to move both directions, read only is fine.
Once you do that, your old Mac should show up in the Finder Sidebar under "Locations." Click on it and log in using the necessary credentials for the user you set up in Sharing.
When you turn on File Sharing, your user Public folder is automatically shared. You could just connect to that and put all the files in there.
You could try to share out your Desktop, Documents, Downloads, but people have posted with problems trying to share those out.
Are you looking for a solution you will use once to get things set up or long-term exchanging of files?