Search for a good institute for a placement oriented course in your region, otherwise best place to search is bangalore, hyderabad or pune.
Many VLSI core jobs lie in the area of physical design, FE/BE design, digital design, Verification etc. Getting a job in FE/Verification is easier than BE/physical. Choice is your to decide the field. For verification you have to be strong in programming like verilog, system verilog, UVM, OVM, VVM etc. For physical design job you need to be strong in core VLSI subjects and need to know the design tools operation. All the Best