You can wear pretty much standard bike kit on a 'bent if you want to. Obviously, back pockets are no use to you any more. Gloves are definitely still A Good Thing, though.
I tend to avoid padded bike shorts, but if your best riding kit involves a pad, it's not going to cause any problems. I tend to prefer sporty knickers and unpadded tights. Or you can wear baggies with regular pants.
The main thing to remember is that waterproofs behave differently on a 'bent. Your back will sweat more (even on a mesh seat) and you could end up with a pool of rainwater in your lap if it crashes it down.