How to buy postpaid sim in USA while traveling in Student exchange program from 5 months

Simple answer is don't.   As a student from another country you are unlikely to be able to pass the credit check required for a postpaid plan.  

Why on Earth would you get a postpaid plan when you can get prepaid for $50 and get unlimited plus data plan 10 GB of hotspot use?

Taxes on prepaid are minimal.

You will need to bring or buy a compatible phone.  Because AT&T is in the process of shutting down its 3-g Network, there are many phones that it absolutely will not allow on its Network. Unless you have an iPhone, you need to check and make sure your phone is acceptable, otherwise you will have to change phones.

Unlocked phones that will work on AT&T now that 3-g is going away

Long distance calling plans can be added to your Prepaid Plan which allow you to call other countries for a set amount each month rather than as a pay-per-use at $3 a minute.

Plans. https://www.att.com/prepaid/https;//www.att.com/prepaid/

Long distance

https://www.att.com/prepaid/international-calling/

Just an FYI, if you plan on going home during school breaks, you should be aware that prepaid probably does not roam in your home country. It is a lot cheaper to switch to pay-per-use plan temporarily or terminate usage entirely when not in  USA.  Postpaid does not reduce charges for being out of the country and not using service.