Automobile Engineering

As international students in the United States you will study on F-1 student visa. The F-1 entitles you to apply for Optional Practical Training (OPT) at the end of your education in USA. This will allows you to stay in the US for up to a year to apply the knowledge you have learned through work related to your major field of study. If you have a degree from the government's designated list of STEM subjects you will probably be eligible to extend your OPT by another 17 months, meaning you can stay in the US for up to 29 months after graduation. The extension is designed to give you time to gain a permanent job offer and sponsorship from an US employer, which allows you to apply for the H1-B, a more long-term visa.

Both mechanical and automobile engineering are good to pursue as career.
Mechanical engineering is the father branch of automobile engineering.
Mechanical deals with the basics and core concepts that are used in almost every mechanical object but automobile deals with only subjects or courses that would help in making or studying an automobile altogether. Basics are same such as thermodynamics, strength of material, theory of machines, heat transfer, refrigeration etc
The better would be to do mechanical engineering as it would help you getting good job opportunities including jobs in automobile sector too.
These are some of the best universities best for pursuing your engineering:
IITs
UPES,
BITS Pilani