I am graduating in mechanical engineering in Dec'15. When should I take the GRE? Should I go to grad school immediately after, or do an internship and go back to school in the Fall?

GRE: You should definitely take the GRE by mid October. That gives you enough time to send your reports in time if you wish to go to grad school in Fall. Giving the GRE early also allows you the liberty of preparing and retaking the test if you decide to apply for Spring admissions. Grad school applications: I would suggest that you do apply to your set list of Schools for the Fall admissions. The only reason you should be looking for an internship is to improve your profile for grad school. Now it would be possible that your profile is already good enough for the schools you are trying to get into. If you fail in your Fall admissions and are enthusiastically intent on going to grad school, then it would be advisable to look for internships. Your chances on getting an internship ironically depends on your profile and your location.

If you are on a roll and can afford it - stay in school until the graduate degree is achieved.  Some schools won't make you take the GRE if you go directly into the grad program, others will require it to ensure you are ready and capable of completing the grad program from the start. Can't speak for any of the colleges/universities for what is available, but you need to speak up for the GTA or GA positions immediately before all the slots are taken once you know you are sticking around for grad school