The important thing to realize about street clothes is that you can often solve comfort problems by either selecting the right clothes or using a simple accessory item.
Legs
I’m mostly looking at this from the perspective of somebody who dresses fairly casual. I really can’t talk about how to commute in a suit and tie because I don’t need to wear one.
The biggest problem for trousers is the ankles. If the trouser legs just rub up against the chain, it’ll look messy and accumulate a mix of oil, dirt, and metal particles. The oil and dirt will wash out, but the metal particles won’t, so you’ll have marks on your pants. It can also get caught, which can result in torn pants and crashes.
Thus, you want to keep the trouser ankles away from the chain. Unless you wear shorts, you’ll need leg bands. Most of them double as reflectors. None of them are perfect. I’ve had leg bands that used a buckle and had a loose section get caught in the chain and broken. The slap-on bands don’t restrain jeans with enough force. The velcro bands work reasonably well, but are far from perfect. I have a broken Planet Bike strap that lights up, but if you wear it on the right leg, it’ll get caught in the chain and broken.
Some of my female cycling friends bike in skirts. This works better than you’d think. If you get a Ladies frame and have a rear rack, you can wear a fairly long dress and it won’t get caught or fly up… pretty much like they did with the feminist revolution when women used bikes to be free to leave home. Or one of my friends rigged up a skirt clip that goes around her leg like a garter to keep the skirt down.
Normal clothes made from technical fabrics
You can buy clothes that look like normal clothing made out of these wicking materials that will make a ride on a hot day a lot more comfortable. Conversely, on a cold day, I found a good under-layer works wonders. I’d wear a long-sleeve technical t-shirt and then I’d wear a normal long-sleeved shirt over it and the combination would let me show up to work dry and comfortable and not too chilly.
Rain
Rain clothes for cycling are hard. See, you might think it’s OK to wear a waterproof rain-suit over your clothes, but this will end up with you wet from your own sweat that the rain-suit is preventing from evaporating.
Thus, if it’s hot and rainy, it’s often better to just wear something you won’t mind getting wet made from a fast-drying material instead of trying to stay dry. Out here, we don’t get hot and rainy, just cold and rainy.
There are fabrics that claim to let water vapor from your sweat through but cause water falling on them, but they are far from perfect. What happens is that you end up a little wet, but not drenched. I’m pretty sure that do end up dryer than you would be in a waterproof outfit.
I tend to prefer to wear water-proof pants while I’m commuting to work in the rain. They are probably running-oriented, but they are also loose enough that if I get caught in a storm, I can wear them on top of my pants. On the other hand, I wear a water-resistant jacket and a shirt that will dry quickly.
You can usually find these for reasonable amounts of money in non-cyclist-oriented form at normal stores.