After acquiring basic required degree in Electronics Engineering or allied Branches or Postgraduate degree in Physics one can choose specialization in this field.
In US:
If you wish to do post graduate degree in US you will have to take up GRE, TOFEL Then follow the required procedure to get into MS and start with MS in VLSI Designing.

To choose colleges offering MS in VLSI designing go to websites like
http://www.campustours.com/
http://www.petersons.com/
http://www.kaplan.com/
And do comparative analysis of schools offering courses in VLSI.
You can search for VLSI and then go to the respective department of grad school to find out the research areas and check if they match to your areas of interests.

In India:
Post Graduate Degree
For the people who have passion for Chip Designing and want to learn it in India you need not get disappointed. Though major job opportunities are available in US nowadays all big companies are opening their offices in India and most of them are located in Bangalore and Hyderabad. So there is good future in this field in India also.
If you wish to pursue post graduate degree here in India you need to take up GATE (conducted by Indian Institute of Technology on second Sunday of every February) in Electrical Engineering/Computer Engineering. Then after clearing GATE you can apply for MTech course. There are different specializations available at each IIT school.
For more details you can go to websites of IIT
IIT Delhi
IIT Bombay
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Guvahati
IIT Rurki

Post Graduate Diploma
Here in India you have one more option to do specialization in this field. But that is not in depth as MS or Mtech but those are Diploma courses offered by CDAC
CDAC offers 2 Diploma courses for Electronics/Electrical Engineers or Engineers of related branches who want to make a career in digital IC designing. The courses are of short duration and not in-depth like Degree courses obliviously but good one to enter in this field.

What Is CDAC?
C-DAC and ACTS
Established in March 1988, as a Scientific Society of the Department of Information Technology, Government of India, the Centre for Development of Advanced Computing (C-DAC), is primarily an R&D institution involved in design, development and deployment of advanced Information Technology (IT) based solutions including a range of high performance parallel computers known by the PARAM series.
C-DAC's Advanced Computing Training School (ACTS) is dedicated to creating high quality manpower in IT through a network of more than 100 Authorized Training Centres (ATC) in India, besides C-DAC's own centres in Pune, Delhi, Hyderabad & Bangalore.

DVLSI: : DIPLOMA IN VLSI DESIGNING

Course Focus
The Diploma in Very Large Scale Integration Design (DVLSI) is a course intended to launch present and future electronic designers into the vast field of Electronic Design Automation. The course contents have been designed keeping in view the emerging trends in the field of VLSI design technology and emerging needs for skilled manpower. For more details go to http://acts.cdacindia.com/vlsimain/vlsimain.htm

DESD : DIPLOMA IN EMBEDDED SYSTEMS DESIGN

Course Focus
This course is more useful for the people who have interest in their respective stream of Engineering like communications, Process control , Consumer Electronics as well as Software . This course gives equal emphasis on hardware and software thus enabling Electronic design Engineers to meet needs of Electronics and IT Industry for the development of the Embedded Systems. For more details go to http://www.cdacindia.com/html/misc/desd/desd.asp

Further Information
For further information you can mail directly to
webmaster@cdacindia.com
acts@cdac.ernet.in
Or visit website www.acts.cdacindia.com