142 West 109th St, New York City

Just after his 20th birthday in August 1981, Barry Obama (and Phil Boerne) transferred from Occidental College in Los Angeles  to Columbia in New York, but the university had no place for him to live, so Obama  had to spent his first night in NYC loitering the streets. Boerne may have already been at this apartment and Obama couldn't get in, but they were living here as room-mates (a two-bedroom / three-story walk-up at 142 W 109th St, # 3E, New York, NY 10025 - Owner of the building: Bita Sassouni).

BY HIS account: When he arrived at the apartment just after 10:00 PM, there was no one at home. He tells of waiting on the front stoop until well past midnight, and, not having enough money to rent a room in a cheap hotel, he crawled through a hole in a fence across the street, found a garbage-strewn alley, made a pallet with his luggage, and went to sleep. He awakened the next morning with a white hen pecking at some garbage near his feet. Debunked? Source: http://www.aim.org/guest-column/the-summer-of-1981/ 

(It's still unclear if they actually traveled to NYC together or just met up there.)

Barry and Phil only spent a brief time together as room-mates, but while they were together they visited museums, ate breakfast at Tom's Restaurant on Broadway, and took walks around Central Park. Sometimes they would have discussions about apartheid in South Africa and go to campus rallies together.

The living arrangement between Barry and Phil  dissolved for various reasons and Obama moved out after his first semester.

See:

Walking tour visits Obama's old NYC haunts 

While still at  142 W 109th, Obama's best friend Siddiqi would come to visit him and Phil.

(Siddiqi is visiting Obama and Phil, not yet living with Obama.)

This short section calls for some speculation:

After Obama moved out, he may have briefly lived in a studio apartment for a few months (at 622 W. 114th St, New York, NY 10025) before moving to the East side because this address has been mentioned elsewhere, though he may only have been staying with a friend or an acquaintance for a short time before living with Siddiqi on the East side.

There was some mention of Obama moving around from place to place. And frankly, this building does appear to look a little upscale and too pricey for Obama's resources at that time; and also Obama had complained about Phil (or the previous landlord) keeping the deposit from the West 109th apartment.

Aside: Phil says that after Obama moved out of West 109th, Obama and some friends would occasionally visit him at his new apartment in Brooklyn Heights (near Park Slope) all during Obama's time in New York before he moved to Chicago. "The last time I last saw him in 1985, he wanted to be a writer."

622 West 114th St, New York

Movie director Cecille B. DeMille lived there from 1906 to 1913.

Obama in NYC

339 East 94th Street, New York City

Later in 1982, Obama lived with his best friend Siddiqi at a sixth-floor walkup apartment  (339 E. 94th Street, # 6A, New York, NY 10128) - (Current owner of the building since 1987: Faith Ministries Inc.) The 1982-83 Columbia University student directory shows Obama living here during his senior year. New York telephone directories show Obama lived there from 1982 to 1985 before he moved to Chicago.

Obama had first met Siddiqi when he attended Occidental College in Los Angeles. Obama had been living with Pakistani students Chandoo and Hamid when Siddiqi was there to visit. (When Obama first arrived at the small liberal arts college nicknamed "Oxy", his freshman roommates were Imad Husain, - a Pakistani - who's now a Boston banker) and Paul Carpenter, now a Los Angeles lawyer.) http://dyn.politico.com/printstory.cfm?uuid=F1C46FDE-3048-5C12-00ED1702A906DC5F 

(*See below: Chandoo and Hamid from Karachi)

At that time (from 1981 to 1984) it would have cost about $450 a month two rent a two-bedroom apartment at 339 E. 94th Street, so says the current long-time residing neighbors.