There are similar answers, but mine is more about Asian sights.
If you are visiting a Buddhist places, temples do not really force male or female to wear specific clothing, but too revealing cloths are frowned upon by locals. If you plan to visit a Buddhist place, try not to wear shorts, bikini, etc. Anything that covers until your knees and covers shoulders are generally accepted. Also, as for Buddhists, people wear white-ish clothing to temples. Try to wear something of white color, but not red, black, etc.
You can easily find something to cover up you outside of such temples.
Also, be careful when you wear T-Shirts with a religious leaders, icons, etc printed. (I'm from Sri Lanka, where it's an offense to print such images on fashion items and inappropriate materiel).
If you are visiting an Islam place, a mosque for example, sometimes there are rules (rather than customs). The national mosque in Malaysia, for example, allows women to visit then place, skirts are allowed, but if the dress is too revealing, they provide you some clothing to cover you (free, you have to return it).
For Hindu places, a dress code is hardly necessary, but generally you wear something that covers your knees and shoulders. If you are too into the Hindu places, do a little research about the corresponding god (most Hindu religious places are dedicated to a specific god: Vishnu, Krishna, etc), do some research about what are the colors relevant to that god. You will be highly appreciated if you wear that color to the place. This is not about obeying their customs, but about mixing up with the locals.
Others have mentioned about Christian places, so I'm not going to mention it here.