In reply to Dan James:
Definitely an ice axe, but you'll also need to learn how to use it. It's not a walking stick, you use it to self-arrest if you slip, and also for belays on more technical ground. If you only intend to do winter hillwalking then a light B-rated walker's axe is probably sufficient, but if you think you might get into scrambles and more difficult terrain then it's probably worth going for a T-rated mountaineering axe, which is stronger. Avoid the technical ice climbing axes though. There's lots of advice on-line on how to choose an axe.
You should also get boots which will take crampons. Gaiters are a personal choice, I like them and wear them a lot throughout the year to keep mud and crap off my trousers, but others dislike them. Some winter trousers have internal gaiters which will keep snow out of your boots.
When wearing crampons, gaiters reduce the chance of catching the points in your trousers (and protect the trousers from crampon tears) but you can still catch them on the straps. Either way you end up on your face, and maybe needing your axe to self-arrest.
It would be well worth going on a winter skills course, where in addition to learning the necessary skills you'll be able to try out gear and discuss what to get with the instructors.