今日、空いてますか？暇ですか？って英語でなんて言うの？

We use this question when we want to know if a person has any free time in his or her normal
schedule.
A work colleague could ask the question in the work place if he needs help with something.
A man would ask a woman this question if he wanted to spend time with her on a first date.

These are friendly ways to as if you can talk.

Hello Yuki :-) I hope this helps!

A) Would you like to meet up sometime ?
-> In this sentence you are leaving the decision in her hands asking her if she would like to meet up sometime soon and then from her reply you will be able to set a date that suits you both!

B) Are you free tomorrow?
-> In this sentence you are asking the girl if she will be available. (Free)

Maybe it's better not to try to be too clever? Just ask her to do something with you without a time reference. You do not want her to be able to say, "No!"
"What would you say to a coffee in town?"
"Well, when were you thinking?"
"Anytime...now, later, tomorrow? What's best for you?'
"Mmmmm tomorrow evening."
"Perfect. I'll pick you up at 6pm."
"OK"

You can ask :

"Are you free today?"
"Are you busy today?"
"Can I see you today?"
"Can we meet up today?"
"Can I visit you today?"
"I'd like to see you today."

It's a good idea to ask if she has plans before asking her out, that way you will know if she's actually busy if she says she is. To keep it casual you could say; what are you up to today? or are you free?

You can ask somebody if he/she is free to to join you for a night out to the movies, to eat out, or to an impromptu party. It really depends on if that person does not have any prior commitments. If he/she has prior commitments, the answer will be negative, but, if he/she does not have any prior commitments, the answer will be positive.
So, you may ask:
Do you have time for an evening out today?
or
Are you available for a night to the movies today?
or
I am in the mood to eat out today. Will you please join me?

Asking people on a date is a lot of pressure, especially because there is always the possibility of getteing rejected. Clearing the path by making sure they are free is a good start to the “date asking.” Asking what’s on their agenda/schedule for the day is a clever way to see if they have anything planned. Sometimes simply asking them if they want to hang/chill/kick it later is good too.