今週、予定ある？って英語でなんて言うの？

If we ask what someone is up to, then we are casually asking what they are doing/will be doing. "Have you been up to much today?" A more formal way to ask would be to ask what someone has planned. "What do you have planned for the summer?"

Another casual to put it, would to simply say "What are you up to this week?". People will understand this as asking about their plans and appointments. "Are you doing anything interesting this week" - This way of asking the questions, steers the conversation towards the good things they have planned for the week. This can be good way to elicit positive responses from people.

Because you are going to be saying this to your family you can be very informal. "What are your plans for the week?" Is a proper and casual way to ask their plans for the upcoming week. "Do you have anything on the go this week?" Is a way to ask is they have any plans or if they are doing anything this week. "What's up with you this week?" Is asking casual "What's up?" but expanding it to ask "What is going on this week."

A common way to ask an open question about 'how busy' someone is in the forthcoming days would bbe to ask them about their 'timetable' or 'schedule'. 'What's your schedule looking like this afternoon?'

"To get up to something" means "to do something". When we say "the week ahead", we mean the week that is about to happen or has already started. So if you ask your family on Monday "What are your plans for the week ahead?", they will know you mean this week and not the next week.

So what are your plans this week? Is a common phrase in English when wanting to enquire about what a person or persons intend to do with their time over the coming week.