I tend to try to not buy used gear that my life may depend on. Crampons and ice axes are included in that. Also, things like carabiners, ropes, and anything else that could have been screwed with by a previous owner. But that's just me. Sitting here thinking about it, there isn't much you can do to screw with an ice axe that wouldn't be readily apparent by just inspecting it, so maybe just avoid buying a used one on eBay, or anywhere that you can't visually inspect it first.
Other than that, expect to buy crampons that will work with your boots, or boots that will work with your crampons. Find an ice axe that fits what you'll be using it for (they're all pretty similar) and that feels good in your hands. It isn't really rocket science.
