With the right bike, your normal clothes will do just fine.
Provided your clothes don’t restrict your movement or aren’t too flappy, you can wear what you’d wear to walk, drive or get the bus. We’re going to work here, not climbing Everest.
If your commute is short, there’s nothing stopping you riding to work, school or wherever in the clothes that you’ve got. Provided you don’t have to be super formal and your bike is equipped with flat pedals and mudguards, you can easily ride to work, do your thing and then ride home all in the same standard, everyday gear.
Really, that's it. If it rains, wear a waterproof. If it's cold, layer up. We could overcomplicate it, but we won't.
Commute longer distances or use your commute as training/gym session?
If so then bike specific clothes might be a better bet. Check out our advice for mile-eaters here.