I like to think of can as an expression of possibility, and may as an expression of probability. There's a major difference between the two words, and hopefully explaining it will help you understand the difference between can and may.
Possibility is binary. Something is either possible or impossible. There's no gray area. You either can or can't do it.
Probability can be strong, weak, and anything in between. It's a percentage, statistical chance for something to happen. You may do it, but you may not do it just as well. We'll find out in the future.
Can also expresses ability. It says you're competent or skilled enough to do it, that you have enough resources to do it, or that you're capable of doing it. May doesn't express ability.
On top of that, both can and may can express permission, They tell you if you're allowed to do something.
Besides a few other minute differences here and there that I'm not going to go into detail about here, that's the basic distinction between the two, unless I forgot to mention something else obvious but important.
Because there's a bit of overlap in meaning between can and may, you can sometimes use them interchangeably. Sometimes, however, you can't do that. It all depends on what meaning you intend to convey. Usually, it's perfectly understandable for the listener what the speaker meant, but, technically, it's all up to the listener's interpretation, which is crucial for some ages old jokes.
A student asks the teacher "Can I go to the toilet?", asking for permission, but because can can also express ability, the teacher might feel like making a bit of fun of the student, and jokingly reply with "I don't know, can you?", which asks if the student knows how to use the toilet all by themselves. The joke would not work if the student used may instead. "May I go to the toilet?" asks about permission, but not about ability.
A child asks their mother "Mom, can I paint the walls black?", and the mom replies with "Of course you can, honey, but you may not!"
A friend of mine, who worked as a call center operator, had to listen to a half-an-hour long tirade about the importance of the difference between "Hello, my name is XXX. How can/may I help you?" because he had deviated from the script by one word, saying "How can I help you?" instead of "How may I help you?"
That being said, "We may go to the Alps next week" sounds like you haven't decided where you will go next week yet, maybe the Alps, maybe somewhere else, maybe nowhere at all. It's also entirely possible that you're just saying that you're allowed to go there because, for instance, you're all minors who can't make such decisions yourselves yet, and you first needed to ask your parents. Now that they've agreed, you may go to the Alps. In isolation, the sentence can mean either of these things, and it's impossible to tell what you mean without more context.
"We can go to the Alps next week" also sounds like you have the permission to go there. It's also entirely possible that you're saying you're capable of doing that. Maybe the trip is expensive and time-consuming, and you want to communicate that you have enough time and money to do that. Again, it's impossible to tell what you mean without more context.
So, with "(...) because we're free", I'd much rather prefer can. Being free sounds like something that will make you capable of going there, and can is appropriate for expressing that idea. I'm curious if the native speakers of this forum agree with my opinion here.
