My A-School is in Great Lakes
 

My A school is in Great Lakes following boot camp and I am wondering since I will be in A school for 16 weeks is there anything to do in Great Lakes on the base while off duty? And am I even aloud to leave base to do stuff if I am off duty?

You'll phase up and earn more privileges.
There's a shopping center with movie theater near by. And a lot of people go into Chicago on the weekends.

As a student, your life will be controlled and micromanaged for the most part. You will always have a liberty buddy regardless what phase, unless you're a phase 3+ (only for fleet returnees or students who have completed A school).
Yes you're allowed leave base during off duty, but you also have a curfew time to be back, even for phase 3. They will give you all that information during indoctrination week. Just be smart and responsible for everything you do. I've heard way too many people get masted early in their career for stupid things and being irresponsible. Great Lakes is okay but I suggest exploring if you have the chance. It's not worth it, wait until you're done with A school and onto your next command. It sucks in great lakes, in my opinion, but it does get better later on if you do what you're supposed to do. There are lots to do on base as well. Just check out the MWR resources.
I'm not a fan of movies and shopping myself. Instead, I waste all of my money on traveling. I was able to go see places from Minnesota to Canada to Missouri and the rest of the midwestern states and still crave for more traveling. I joined the navy to see the world, and there's definitely lots to see. Good luck.