You can incorporate a coaching institute in Patna just like how you would incorporate any other general business-
Register for sole proprietorship or a Pvt Ltd firm.
Get shops and establishment license
Obtain trade license
Open a current account
Get you GST & and other taxation paperwork done.
It's good to engage a company registration and taxation consultant. This gives you more time to execute your business expansion plans. Feel free to reach out to our team for help.