First of all this question does not have any technical ground on category Mechanical to be asked at enggforum ,but enggforum - community is committed towards building skills among students.
Regarding your question :
Its amazing to see your vision on future plans.You should pursue your higher studies (M.S).
Internships in a good company would really add up to your resumes and experience.
Internships are available in various dimensions say for eg: online , offline or regular basis.If it is a regular one then your institution must allow know.
Also companies differ in their terms in selecting students i.e pre-final / final year students.Hence i would suggest you to get start to search relevant internships and get an experience in some good companies.
GRE : Apart from your regular studies and internship hunt ,you must spare time for GRE on a daily basis.
Whats the right time to start ?
Get started right now.No time to waste.. The right time is now; so do it now!
All the Best :)