I’m getting stationed at Fort Hood. Will they send me home to get personal belongings? If so, will they make me take my leave days?

You should be receiving a complete briefing from your leadership that will lay out all the "what's next" steps after AIT. This should include timeliines already mentioned in this thread.

Check your orders and DA31. You should get 10 days of leave. I suggest that you not take the whole time. Save as many days of leave as possible in case an emergency comes up and you need to take leave.

you may also be able to have your personal belongings shipped without you physically going home to get them. Go see the transportation office when you get to hood if you don't have them by then.