Every doctor will have a different approach to examining chest X-rays. A commonly used mnemonic for what to look for on a chest X-ray is: It May Prove Quite Right (but) Stop And Be Certain How Lungs Appear:

I = Identification (name, age, sex, indication for X-ray)
M = Markers (differentiate left from right - diagnose dextrocardia)
P = Position - the spinous process of T4 should be between the heads of the clavicle (if it isn't the body is rotated)
Q = Quality - is the film penetrated properly. In a properly penetrated film the vertebral interspaces should be visible behind the central (cardiac) shadow
R = Respiration - chest X-rays are typically done with full inspiration
(but)
S = Soft tissue - look for subcutaneous emphysema (suggestive of trauma), soft tissue swelling
A = Abdomen - look for free abdominal air (suggests penetrating trauma, peritonitis, or recent surgery)
B = Bone - look for fractures (these tend to be at the lateral aspects because of the mechanics - bending moment largest at lateral aspect)
C = Central shadow (cardiac silhouette) - greater than 50% of lateral distance in frontal view at the diaphragm suggests cardiac enlargement (usually secondary to heart failure) or a pericardial effusion). A widened mediastinum may suggest aortic dissection.
H = Hila (of the lungs) - can be affected in lung disease, malignant processes and infection (hilar lymphadenopathy).
L = Lungs - for consolidation, interstitial lung disease (reticular, nodular or reticulonodular), honeycombing, miliary pattern, granulomas, lung masses
A = Absent structures/Apices of the lung (for pneumothorax)