Congrats on the job!
Most consulting firms have a very well-structured on-boarding process, and I am sure that McKinsey does. On day one you will do a lot of the administrative work. You will receive your laptop, your phone, office key badge, your business cards. For me, these things were waiting on my desk, along with a small welcome gift on my desk (back then, we still had individual desks).
You'll get a tour of the office, learn the most important ins and outs (IT support, travel stuff, conferencing etc.) and most likely have a call / meeting / lunch with your Mentor and maybe with the Engagement Manager or partner on your first project.
Then it is most likely off to training or to your first project.
But Guennael is right: Everybody will be nice, everybody will offer their help and give you a warm welcome.
PS: I have also seen new hires (seniors, though) that were immediately shipped to the client and had their stuff sent there by courier. That's a lot less romantic ;-)
PPS: The most important people - be it in your office or at the client's - to have in your team and have friendly relations with are the assistants and then IT support. They can work miracles for you if they like you. So be extra-super-nice to them, especially on your first day. Before you know any consultant's name, learn theirs. It's a well-worn, but true saying: There's no second chance for a first impression.