The typical Greek wardrobe of the Classical period distinguishes itself greatly from the Minoan and Mycenaean dress code and its rugged, exacting geometrical and line-based garments.
The Classical Greeks lived in a different world. The arts were encouraged, even challenged, to create new things. The love for knowledge was greatly embraced and change was everywhere. The collective feeling of that generation is evident in the way that they manifested themselves: freely, elegantly, and softly. This is ironic because this is also a particular time when complex ideas and new discoveries challenged the status quo; in other words, even under a myriad of transformations,the Classical Greeks felt right at home.
Classical clothing manifested the overall psyche of this generation. Men and women chose common sense over excess. Their garments were similar, limited in terms of add-ons (unlike, for instance, the Egyptians). Wool (for cold weather) and linen (for warm weather) were the materials of choice for both genders. The idea of a silhouette was to enhance natural grace rather than to show off the human body. This is why in the classical period vases you see the drapery covering the body rather than exposing it. Softness and subtlety are key. In all, the design is so basic that it is continuously replicated to exactness in Hollywood sets.
The two most important pieces of clothing for both genders were the tunic and a cloak (himation). The tunic, also known as a peplos, or a chiton, was basically a wool or linen long and rectangular cut. It would be folded on the edge so that the upper part of the folding would fall right by the midsection of the body. That fold is called apoptygma. It had holes for the arms and it would be pinned at the shoulder.
Then comes the chiton which is that quintessential robe that hangs diagonally on top of the tunic and cloak.This is of a lighter material than the robe. You could say that the robe was wool and the chiton was linen, to be exact. Everything is supposed to go all the way down to the floor and would be pulled up using a kolpos, or a little extra pouch.
The main difference is that the men's chiton was way shorter than the females. The men's chiton is called an exomis. Another difference is that males actually wore hats sometimes! It is a 'petasos' or a brimmed hat. Even women would war hats but theirs had a bowler hat look. The statue at the bottom of the answer represents Hermes wearing a Petasos.
The shoes for men and women were mainly sandals, boots and shoes that were often used only when out of the home.
The importance of Archaic-Classical Greek fashion is that, although is simple enough to be replicated over and over (even college students can replicate a typical Greek costume for "Greek Night") it is considered one of the most elegant, sophisticated and representative of its historical period. It is literally the time stamp of a golden era.