Your veterinarian will be able to tell you a lot about your dog’s breathing issue from physical examination alone.
Observing your dog’s breathing pattern and effort (either in person or from a video you provide) will give them a lot of information. They will look at your dog’s gum colour, feel around their neck/airway, check their temperature, and most importantly, have a good long listen with their stethoscope (auscultation).
They may also place a probe on your dog’s ear or tongue to assess your dog’s oxygen levels. Based on all of the above, they will be able to let you know how serious/urgent the problem is, and narrow down the issue to one or a handful of possible causes.
After these basic tests, they may recommend further diagnostics; most commonly, your vet will recommend chest x-rays. This allows for a detailed look at the lung fields, and can also get a look at the trachea, and the silhouette of the heart and diaphragm.
In some cases, an ultrasound is recommended to better assess the heart or to quickly check for fluid around the lungs (pleural effusion).
In the case of more complicated or chronic issues, blood and urine tests may be recommended, as well as more advanced imaging like scoping of the nose or lungs (rhinoscopy, bronchoscopy).
How quickly your vet moves along this diagnostic pathway will depend on how urgent the issue is. Fortunately, in most cases, your veterinarian can gain a lot of information from just their physical exam and x-rays.