You already know that you need to listen more, because you are asking the question. Sadly there is no magic words that will make you do that; you just need to learn and believe in the benefits of you doing so.
Sometimes there is a case for giving direct advice. But it should always be presented as such - advice, and not the final word. Everybody should make their own decisions and accept responsibility for them.
Most of the time, the desirable way to help someone with a problem or a question is to use counselling skills - that is to ask helpful questions which help them answer their own question or draw their own conclusion.
First of all, you need to listen and don't interrupt. It can help to make some conscious sign that you are listening - maybe some nods, or maybe even some listening noises (like "mmm" or "right") but don't over use noises and certainly don't use them as a crowbar to stop her talking and get your point in.
In the example you gave:
Alice: "I think I should move to this better school, but I'm worried
I'll miss my old friends".
You: "I'd miss my friends too"
I'd say this was the wrong way. First of all you have given your opinion based on how you would feel. That might not be how she feels. She may now question her own feelings and feel even more confused; or she may follow your advice and then later regret doing so.
But if you use a question, for example:
Alice: "I think I should move to this better school, but I'm worried
I'll miss my old friends".
You: "Which is more important to you?"
... you are inviting her to think for herself about which is more important. This is beneficial for many reasons. First of all, she makes her own decision. This is character building and she won't (or shouldn't) blame anyone else for advising her to take one path over another. But it isn't just about being free from blame... let's say the answer IS her education and she chooses to move - when she thinks back on the decision she should hopefully remember why she made the decision and this reinforces her belief that she made the right decision. She ought to feel more confident that she made the right decision, even when she misses her friends which she inevitably will.
In dealing with complex problems it can also be good to repeat back key points after you have listened and before you follow up with a response.
You need to believe that this is often better than giving advice. If you believe it, you will do it.