How to roam the streets of NYC on your own, explore and visit?

I am living in NYC, but I'm free over the summer - I was wondering if anyone had the best suggestion as to roaming NYC - I mean, I don't just wanna see Times Square or The Statue, or take the bus tours - I want to actually roam the streets and see everything that is culturally significant, and I have 2 months to do it. Local residents might have some idea how to do it - I would appreciate it if you could give me something like a 2-month itinerary. As for starters, I live near Columbia University.