Stuck "Enter your Apple ID password"
From the screen image, I would try a couple of things since you should be able to log in to the Mac and do things other than iCloud. Press cancel in the dialog and then check the date/time on the Mac. It should be within 2 minutes of reality if possible.

Then go to Safari and check at https://icloud.com to be sure you can log in and the account isn't locked or having issues.

At that point, you might need to clean out your keychain or try again in the iCloud preference pane. Worst case, you might need to get on the phone/web chat with iCloud support if it's a problem with your account or Apple's servers.