When we are introduced to our protagonist, Roger Thornhill (played by Cary Grant) he is walking the midst of a busy crowd, establishing the "everyman" nature that makes him so identifiable, as well as vulnerable.
Roger is an advertising executive, or as he puts it, "I lie for a living", (a line that was no doubt influential on the creation of "Mad Men") This is further evidenced moments later when Roger takes a taxi from another man, saying that he has to take his sick secretary to the hospital, when nothing of the sort is happening.
Since Roger is a businessman he is wearing the expected attire of a man of that position in the 1950s, a gray flannel suit, the most popular fashion choice for men at the time. So popular in fact that everyone was wearing them. Thus we get the first hint of the theme of this film. This could be any man, even you. Is it any wonder that the plot hinges on him being mistaken for someone else?
This color choice also has a double meaning, Roger wears gray because of his moral ambiguity, as he has to lie in order to get out of trouble so that he can expose the truth.
The plot thickens when Roger is framed for a murder. He manages to escape custody, and flees to try and clear his name. Despite his picture being in every newspaper in the country, he continues to wear the same clothes, (It may not make sense in retrospect, but it is done to keep the character consistent) although he does don a pair of sunglasses that amusingly fool absolutely no one.
The one time in the film that Roger deliberately alters his appearance is while escaping from a train. He does the classic bit of knocking out a man and stealing his clothes. However, they're not just any clothes, he chooses the uniform of a station redcap, somebody who is one many anonymous workers at the train station, only identified by their eponymous hats. This frustrates the pursuing police officers who knock off the caps one by one in a failed search for their man. This continues the theme of the everyman who easily blends into the crowd.
Towards the end of the film, Roger is shot. For all intents and purposes after this, Roger has "died". His name is cleared with the authorities and they are preparing move in on the real villains.
However, this is when there is a distinct change in Roger's character, which is represented by his change of clothes. While recovering from his gunshot wound at the hospital, he is given a new white shirt. Roger has been reborn and his heroic colored apparel has also inspires him to become a proactive figure. No long content to hide, sneak about, he takes charge by going out to rescue his love interest from the clutches of the bad guy's fortress like a classic white knight.