When PCSing to your first duty station from AIT, what moving support do you get for you and your family?

There will be some time between your graduation and your report date, usually around 45 days. It won't say it on your orders, but you are authorized to report up to 30 days early. Unless, you're going OCONUS, then it's an entirely different conversation.
If possible, I recommend PCSing to your duty station, securing housing, and then taking leave before bringing them out. That's not always an option, but it's easier than shoving a whole family into post lodging for a week or longer while you look for a house.

A couple of good answers already, BUT: each soldier's situation is different. How critical is the Army's need for your MOS in the gaining unit? Is family travel authorized to the new duty station? You'll get more information near the end of your AIT and when you know your assignment.

Generally speaking (every rule has exceptions) the Army will pay people to move you. This includes all of the packing and a limited amount of unpacking. Even if you are unable to make it home for whatever reason, your family shouldn't be stuck trying to do it on their own.

You didn't ask, but here is some unsolicited advice gained from 8 PCSs... Watch the movers like a hawk. Make sure they are properly packing things. Make sure they inventory EVERYTHING with its OWN sticker. Verify the inventory sheets - NOTHING goes on that truck without being listed. (Don't let them tell you that they will put the sticker on later or write it on the sheet later.) Declare anything even remotely valuable as high-value. This includes things like coin collections, electronics (even older ones), baseball cards, and heirlooms, even if the value is only sentimental. Make sure that any hardware is kept with / taped to the furniture it goes with. They LOVE to do a "parts box" and then on the receiving end, no one can figure out which screws go to what.
On the receiving end, make sure you do a thorough inspection of all your stuff within the given timeframe and file your claim promptly.

Bottom line is that if you PCS enough times, some of your stuff WILL get lost or broken, but if you pay attention, LESS of your stuff will be lost or broken, and you will be properly compensated for the stuff that is.