Would it be best to try and earn an associates degree while attending AIT or wait until after AIT graduation?

Not that you are incapable of multitasking, but...go to AIT and learn as much as you can about your J.O.B. Then, in your spare time work on that degree.
Keep up the motivation.

Dont do college. Wait until you get to your duty station. Print out your transcripts and submit a copy to whatever college you will go to, you should get credits for courses you have taken. Also whenever you complete college courses, you can submit them for promotion points, when the time comes. Unless, of course, they changed the requirements. Its been almost a decade since ive had to know that stuff.

Learn your MOS to the fullest.
If you do great in school you might be lucky enough to gain rank, if you are busy doing anything else you won't learn the best of your MOS.
I even spent some weekends studying my TMs and FMs to gain better learning of my MOS and got two promotions during my 6 month AIT.

Focus on AIT. Won't have time to be successful at both.

While I have a great appreciation for your desire to improve yourself, it needs to be put in your pocket for now. Notice I didn’t say loose it... Way too many folks do. Just put it away for the moment and focus on the task at hand. You will have more than enough time down the line to pursue your degree. Just don’t forget this desire to improve when you hit your first duty station.

I honestly doubt you will have the free time needed to knock out all the hoops and ladders to get your AD TA submitted/approved and work on your school work. You are a grown man and can do what you want but you most likely will run into quite a few obstacles trying to do this while at AIT.

I know the Intelligence School house has a relationship with the local community college, and 35 series MOS's can earn 20 semester units for completion of Intel School with units in Intelligence Operations. What easier way to get units for what you are doing. Also, those soldiers that have general education completed can transfer those units to the community college and they will issue an AA degree in Intel Operations. So basically, all SPC's with degrees already could get another degree for the $50 admin fee to the school.

Maybe that relationship exists in your MOS. That would really be your only option. Whatever time you think you might have is going to be taken away from you because, well, you are in the Army, and it's still "basic training" mentality.

You will not have time to do both. You are going to be hard put to learn your MOS, despite any thoughts on the matter otherwise. And your MOS should be your priority. Your continuing education is a secondary and should be started only after your completion of your MOS training.

AIT will take all your time during the day and some evenings for studying. If you try to do college classes when not in AIT classes, this will leave you with little to no personal time. One or the other will sacrifice and considering if you do poorly in AIT you lose a job, I would not try to do both.