Just stumbled across this thread. It sounds to me like you're going to AIT for exactly what I was in the Army as... intel.
You can't rely on the timing nor the possibility of having time off. Delays happen every day in the Army which can result in a longer stint at Leonardwood.
At Goodfellow, the first few months of your time there will be spent in the barracks with authorization to only go to the BX (base exchange) in your uniform. After that time, you'll be able to wear civvies, and eventually you will be permitted to go other places, even off base. That is IF your drills don't decide to put you or your whole company on C-Pass which is essentially you being grounded to your barracks. You won't even be allowed a vehicle until a few months into your stint at GF.
I would absolutely advise you NOT to bring your horse with you to Goodfellow. Reason being, if you get put on C-Pass (which you can be put on that for something as simple as a failed test or your class acted like fools), you will not be able to go take care of your horse. You would have to pay to have him taken care of/ridden.
You need to spend those months focusing on the workload, and trust me, you will need extra study time (everyone needed it!) and you will be stressed to the max. Adding the burden of taking care of your horse will not help at all, and can hinder your training. It is a distraction that is simply not worth it at the time.
Wait until you get to your permanent duty station, and then get your horse out there.
I don't mean to be a downer, but I absolutely know how basic and AIT goes, especially at Leonardwood and Goodfellow. Those are the locations I attended training. If you have any questions, I am here to answer them as best I can!! :)