Cycling shorts are constructed of 6-8 separate panels, so that they not only conform to your body, but move properly during pedaling, without bunching or chafing. They also have padding and a wicking component, which removes moisture from your skin. If you have modesty issues, you can get shorts with a baggy outer shell. For all but the shortest rides, bike-specific shorts will provide more comfort than just about any other type of short.
Jerseys don't need to be skin tight with team logos. A basic jersey will have some pockets in the back for your energy bars or gels, and maybe a phone. The front usually has a short zipper to help a bit with temperature control. And they are made of materials that wick the moisture from your skin, keeping you cool.