If winter walking I always carry my ice axe and crampons. I particularly like to go up Helvelyn in the winter, a great walk any time but even more magical in winter, I am yet to see the tarn frozen, which would make a great photo.
I would recommend, always get to learn how to use them properly first, and would say getting trained on a course, is much better than a book, and then practice what you have learned/been taught, so you don't deskill in it. You never know when it might come in handy. There are many advantages of a course, apart from having your technique criticised and fine tuned, on the 2 day course I went on which was held in Aviemore, we were taught how to read the snow, dangers of drifts caused by wind and how to make a snow hole. Plus it was a fun experience with like minded people.