If you're doing a biking workout indoors, weather is not a factor. Instead, the type of exercise clothes that you wear depend on how hard you want to work out and how much you want to sweat. If you are just pedaling steadily for half an hour while you read a magazine, any shirt, long or short sleeves, and any style of athletic shorts or pants will do. Even casual sports clothing styles are fine for moderate stationary biking.
But if you're taking a Spinning® class, or doing a tempo ride, or just pedaling hard enough to get sweaty, you'll be more comfortable in a moisture-wicking top and shorts or pants.