I still say the best thing to do is start smaller, on well traveled trails with low chance of needing self arrest skills and then work your way up. I bought an ice axe early on and sort of jumped right in. I did take an ice axe and crampon course with the GMC where a class of 12 people ended up being a class of 2 with 1 instructor so we got a lot of personal attention and practice. My self arrest skills have improved with every winter mountaineering class I have taken and I will continue to practice and carry my axe with me.
I think it is personal preference. I usually just have it on my pack and only take it out to use in a situation where I feel I might need it (remember it does no good just strapped to your pack). To name a few of these situations off the top of my head . . .
1. The trip from Lakes of the Clouds to the summit of Mount Washington in March.
2. The short section near the summit of Mount Jefferson hiking in from Thunderstorm Junction, also in March and
3. Up the Saddle Trail from Chimney Pond on our way to Baxter summit in February.
I think it is important to learn how to use it and when to use it. Most importantly remember that everyone has an opinion and they almost never agree 100% . . . you need to figure out your comfort levels and work within those . . . IF you think an ice axe will give you a false sense of security and make you do something you shouldn't be doing then don't buy one BUT I personally think it is a good idea to buy one, learn to use it, carry it always in winter and USE it when NEEDED. I see no need to limit gear because you cannot make a good clear decision about your risk and experience level.
I have only once left my ice axe and crampons in the car so I wouldn't go into a situation that would prove to be dangerous and that was on Long's Peak in Colorado in June and I was solo and I didn't want to tempt fate and my desire to make summit under conditions that might have put me in over my head BUT in general, I am a big proponent of carrying gear that might be needed and CHOOSE not to use it and turn around rather than find myself further along in a situation without the needed gear.
I know all this confusing to someone just starting out, I was at that same place myself not long ago BUT if you start slow and remember that turning around BEFORE you get in trouble is the BEST "gear" you can carry then you will find yourself deciding for yourself what is best.
Good Luck and remember to have fun
sli74