I don't bike often, but when I do, I have a one-mile bike to the subway and then a mile on the other side. It's fine on this side of the bay where it's cool and foggy in the morning, but even the few minutes on the other side leaves me a bit sweaty. On days I bike, I just try to wear very practical clothes and shoes, and sometimes I change my tshirt. I don't wear makeup, but hair is an issue - my curls are rather fragile so if I wear my helmet I just resign myself to wearing my hair up or in a ponytail for the day.
To avoid sweat I never wear a jacket on my way to work, even if it's chilly. I often wear sleeveless blouses then wash my armpits at work. (I don't use antiperspirant, but Tom's of Maine deodorant is amazingly effective for me.)
I have a bike-friendly backpack that's big enough to hold a jacket, a change of tee, etc. Evening is always cold at all times of year, so I have to remember to bring warm clothes for the return trip. I'd get panniers but I don't bike often enough to want to spend money on it.
I used to bike ~3 miles to the Caltrain station, then another mile to the Stanford campus, where it's much hotter and sunnier. That also wasn't a real problem for sweat. But in either case, because I hate feeling hot, I sometimes walk into the cold room (4C / 40F) immediately upon arriving at lab!