One of my all-time favorite weekend activities is the “coffee shop crawl.” Saturday or Sunday morning, ideally after a nice run, Caroline and I like to spend several hours at a string of coffee shops. It's like a pub-crawl, only nerdier. She’ll read the paper or work and I’ll usually be catching up on blogs, the news, or reading a book. I usually get chai, iced tea, or a small snack at each place we stop since I like to avoid coffee on the weekends.
While she has a weakness for Starbucks Frappuccinos, I prefer the the more pretentious independent or boutique shops. Some of my favorites in NYC: Third Rail, Pushcart, Gimmie (from my hometown Ithaca, NY), Stumptown, Local, McNally Jackson bookstore/coffee shop and Prodigy.
The routine begins when I pull up Yelp or Foursquare in our neighborhood, or one we want to check out for a few hours, and find 2-3 places we want to stop at that look good (wifi, seating w tables, bathrooms, good reviews). This turns out to be a GREAT way to explore different parts of NYC, catch up on reading, spend time with Caroline, and unwind after travel or a long week.
One suggested tip: when traveling to a new city for the first time I use Yelp or Foursquare to find the highest rated espresso in the city and head to that area. It's usually one of the most interesting neighborhoods in the city, not fully gentrified, and filled with cool shops, bars, and restaurants.