There is no save button. How can I tell if the Trello card is saved or not?
Most things on a card will have a "Save" button. In addition to that, other items, such as editing labels or assigning members, will display a tick or checkmark (✔) next to the action you've just completed.
For the most part, when the card content changes, it will autosave.
Sometimes, if you hit ESC on a field, or close it before it has a chance to save, you will see a note like this:
You have unsaved edits on this field. View edits or discard.
You can then make sure your edits are saved to the card before moving onto your next task.