If you are studying in class 12, then JEE Main and JEE Advanced will be required to study petroleum engineering at some of the best colleges in the country. IIPE in Vizag and RGIPT are exclusive institutes for PE and the entry is through JEE advanced. IIT Dhanbad also offers it and again entry is through JEE Advanced. UPES Dehradun is a private university that offers this and PDPU is another option to study the branch. To make a career in petroleum engineering, you can also opt for Chemical engineering as most of these institutes also offer chemical engineering and you could apply for jobs in PE even if you do chemical engineering. For Chemical engineering, you will have many options as many IITs, NITs also offer it. 

You could apply to the oil companies after competing your B.Tech or even appear for GATE for recruitment in some PSUs.