You want an axe that looks like this:
And not like this:
Because you're mostly walking with it, not climbing vertical ice. Get someone to teach you how to self-arrest, and practice somewhere with a safe run-out until you can do it without thinking.
I'd recommend 12-point rather than 10-point crampons unless you're really light, it helps spread the load better. And learn how to use them on easy terrain before steep stuff too - for most people the biggest thing to learn is to bend your ankle and put the points flat against the ice/snow rather than trying to kick edges like a boot. Everyone walks like a constipated cowboy at first, otherwise you swing too close to your leg with each step and end up slicing your gaiters/legs or catching a point and tripping yourself up.
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