Yes you will be issued BCGs while in Basic Training. You can pretty much wear anything that you want as long as they comforn to AR 670-1. The Army will issue two sets of BCGs and one set of more modern looking frames each year of which you have several to choose from once you get to your unit.
I order my glasses from Zenni Optical
They are inexpensive and easily replaced. I have been using the simple $8 basic frame for the last several years.
Glasses while deployed and in the Infantry. It is highly recommended that if you do not get issued a set of Prescription ballistic glasses. By that I am not referring to the ballistic eyepro with the insert-able lenses. Order a pair or Two of the Prescription Ballistic Wiley X goggles. You won't regret it. They are a couple hundred bucks a pair but well worth it.
I am not serving in the infantry any more but can tell you that wearing the glasses that are issued to you with the interchangeable lenses is a pain in the 5th point of contact. The Army ought to issue prescription eyepro.
Do not under any circumstances attempt to wear contact lenses at any time while at basic training, in the field, or while deployed.
Looking in to having Lasic done is something you should think about as well.