When is TA available to soldiers? I.e: how many months of service until I can access TA for schooling, or is it just after AIT ends?

there used to be a 1 year wait, now you can request TA when you arrive at your unit. Get with your NCO to help you set up a GoArmyEd account

As I stated in your previous question, you are not authorized TA until you get to your unit. You will inprocess the Education center and then you can access TA

Immediately after completing AIT. You must get counseling from your ed center, pick your classes and submit the proper aplication to your permanent party commander. Once approved, you can begin classes, up to the value of $4500 per semester.