How do I keep the ice skates from hurting my ankles and cutting them?

If it's rental skates, they're bound to be uncomfortable in one way or another. They don't have any padding.

Are the laces tied too tightly? You should be able to fit one finger between the skate and your shin after tying the laces. If that doesn't fix the problem, try gel Bunga sleeve pads. They're a gel sleeve that slips over your ankle and cushions the place where the skate cuts off.

If you're breaking in a new skate, you're bound to get lacebite from it. Only tie the bottom three hooks and just bear with it for a few weeks.

Long socks just rub you and don't help much. Snug tights will work better, or tuck in the top of your footless tights / leggings.

Long socks won't really help... I'd say just stick to regular tights. I think there might be a bunga pad/gellies for your Achilles area, but don't quote me on that... you could look, though, it may be worth it. Other causes could be rentals (they are worthless), you're tying them too tight (if you get a good pair of skates with a flexible synthetic/leather piece on the back there by your Achilles it shouldn't really bother you), they need to be broken in (some people suggest not tying up the top notch when breaking in skates), the skates just fit you wrong/don't have that padding at the back... If none of those really ring a bell, maybe ask your coach...?

It depends on the type of skate you are using. Figure our hockey? If you're using figure then I suggest following the other ansewer advice.

If you are using hockey skates then it's a little differnet. If you are, then don't loosen your skates. You want to keep them tight, untill you have strong leg muscles to keep the up and not tilted in or out.

Try a pair of thicker socks, which might help. Or go to a resale store and buy a pair of skates. You will find cheep, broken in skates, that are usually really good.