Well I always like going straight to the good stuff when I can. No sense in buying something semi good quality to then find out you want to buy the nicer one. With ice axes it depends on your style. Mine is pretty modern and light weight but did ok on Mount Hood, but ironically if I had my friends heavy ice axe that is 30 years or so old it would have been even better because the spike was sharper and the planting was powerful. But most the time lighter weight is better. Winter personally is my favorite season (except when avalanche danger is so high I cannot enjoy it). Getting to use an ice axe I found was one of the biggest steps in mountaineering, easy to learn but super important. Once I knew how to use that axe, man I had my desires for peaks go though the roof, and the doors of opportunity opened up.
Anyways as for crampons, I played the cheap route and ended up with crampons so dull that on ice it becomes a bit sketchy because the crampons did not go in (they were rounded and very short). They still work great like micro spikes, but bad on blue ice. So if you become a winter lover like me, get a decent pair. And if your trying to save money, buy them used. A lot of my climbing gear is used, and heck it's very good quality. Now I just need to get better gloves.