You have many great answers here but I think I could provide you with a lot more information specific to you.
First of all, I wouldn't advise that you do not apply for an internship. Instead, I will suggest that you apply for the real job -- go for the associate role. But only do this when you are ready. And by ready, I mean you are ready for a case interview. You do have many great resources here but I'm afraid many coaches here don't deal with candidates of your kind. If they do, I'm not convinced (with all due apology and no pun intended to anyone) they can address many of your deepest fears and myths as a PhD candidate; which is why I will recommend The Consulting Offer 1 by Firmsconsulting. The Star candidate in that session is Felix, a PhD Candidate currently at Mckinsey.
Second of all, you do have a lot of work to do and if you are currently in your PhD program, I will suggest you start now. Please, do not start to network until you have done your resume and truly understand what consulting wants from candidates like you (again, this is where The Consulting Offer 1 works incredibly well). I have used the service and still using and I am excited that this will work for you.
Thank me later
If you do have any more question, I'd be happy to answer them.