What do you actually wear over it on the way there?

Trying to sort out the practical half of an evening and this is the bit that defeats me. I do not drive, so getting to anything means a bus and a fifteen minute walk.

Do people go dressed with something over the top, or does everyone change when they get there? If it is a coat, what sort of coat, because most of mine are wool and I suspect that is exactly the wrong material against a polished piece.

Also, honestly, how much do you worry about being seen on the way?