Research Greek art. Begin with art history textbooks and check out their bibliographies and "Suggested Readings" sections concerning Greek art and architecture. Study and memorize the art historical vocabulary: know the three Greek architectural orders and the names for the various parts of Greek temples.
Learn about Greek culture. Study Greek mythology so that you can identify gods and understand much of the subject matter of Greek sculpture, architecture and art. For example, the Parthenon of Athens was dedicated to the goddess Athena, and much of the Parthenon's sculptural decoration concerns Athena, including the pediment sculptures, the interior Ionic frieze and the cult statue (now lost) of Athena.
Understanding Greek culture and philosophy will bring insights into the values and standards that Greek art and architecture expresses. The Greeks strongly believed in the balance of oppositional forces, like light and dark, chaos vs. cosmos, reason vs. irrationality, civilization vs. barbarism. Many of the relief and pediment sculptures express myths whose morals have to do with this idea of balance--just look at the metopes of the Parthenon.
Learn about the history of ancient Greece. Understand the divisions between the Archaic, Classical and Hellenistic periods, and the typical aesthetic ideas of each. For example, one of the main differences between the Archaic and Classical periods is the development of "contrapposto" (an Italian word meaning "counterpoise") in the early Classical period.
Archaic sculptures of men and women are stiff: Mostly, they face straight ahead with one foot directly in front of the other and hips parallel to the ground. Classical statuary, on the other hand, appear much more fluid, the figures placing their weight on one leg with the other knee bent and the hips slanted--just like how humans tend to stand comfortably.
Develop an eye. Look at as many examples of Greek art and architecture as possible. Understand the proportions of Greek architecture and learn to see how mathematical ideals worked, for example, in Doric and Ionic temples.
Know that the importance of proportions also emerged in Greek sculpture. One of the most famous Classical Greek statues is the Canon of Polykleitos, who said that "perfection comes about little by little [para mikron] through many numbers." Polykleitos meant that the ideal human form and ideal work of art should be composed of distinct parts, each related to each other as well as the whole statue through mathematical proportions (finger to hand, hand to wrist, hand to arm, arm to leg and so on).
Many believe that Polykleitos' ratios emerged from Pythagorea's ideas on the perfect intervals of the musical scale. Certainly the Greeks believed in the importance of music, math and art, and saw these fields as deeply related.
Understand these proportions, and begin looking for them, not only in Greek art and architecture but also in creations by artists of the Renaissance and Neoclassical periods. Brunelleschi's work, for example--foundational to the beginnings of the Renaissance--continued Greco-Roman ideas of perfect mathematical shapes and proportions.