In 5e, is the breath weapon of a Dragonborn an attack action or can it be combined with another attack? The Player's Handbook does not specify.
A dragonborn's Breath Weapon is an Action to use:
Breath Weapon: You can use your action to exhale destructive energy. Your draconic ancestry determines the size, shape, and damage type of the exhalation.
In 5e there are no swift or movement actions. The "fast" action types are Bonus Action and Reaction. (Movement is not an action type at all.) So the breath weapon is just an Action, which is the rough 5e equivalent of the "standard action" you might be used to from other editions.