I had thought that they checked vaccines when going landside but not going to connections, but as I live in Madrid I've actually never connected there. But regardless they just scan a QR code which, with a US card, you would get after filling out the Spain Travel Health form. The whole process is less than 30 seconds.

The one main tip for people is that, especially after passport control, use the lift rather than the escalator to get down to the train. That can save a minute or two and be the difference between catching one train or waiting for the next depending on timing. It also leaves you right by the best door to get out quickly and be at the front when getting down.