Graduate in May, start work the following January- how to spend the gap time?

There's certainly room to disagree with my take, but this is just me. It sounds like you don't know what you would want to do with the time even if you had it. If you don't know what you want to do right now, you might as well get paid while figuring it out. You're worried about not knowing people from the city where you're going to work, but one of the best ways of doing that is beginning your work in the city and meeting people. Sometimes your dread of doing stuff will be more painful than actually doing anything, so why not start? You can't dread something that you're actively doing; you can only hate it (little consolation there).

I'm hyper-frugal about everything, so I wouldn't have as much experience with this, but as far as affording travel, that's what credit cards are for. Get something with a low introductory rate and let the MBB pay do the heavy lifting. Possibly sounds like a recipe for failure if the pay doesn't come in like you want, but there's something to be said for the joie de vivre of the idea.

But I'd choose the earliest date because I'm a boring curmudgeon.