Tokyo doesn't have a lot of recreational spots, but it has one gem. New York has Central Park, San Francisco has Golden Gate Park, and the residents of Japan's biggest city head to Yoyogi when they want some green space.
There's a lot more to Yoyogi Park, though it's mostly paths with fields and trees. And Tokyo doesn't have a lot of space like that. There are landscaped gardens throughout the town, but it's difficult to find a little green space to sit under a tree and enjoy a sandwich with friends and family.
The best way to get to Yoyogi Park, in my mind, is from Shibuya, after walking through the shopping insanity you find there. Then you can make your way through the promenade and around the park as you wish.
If the temptation for shopping in Shibuya is too great, Harajuku and Yoyogikoen Stations are on either side of the park. I recommend starting from Yoyogikoen, and being sure to swing south to the promenade and on to Shibuya after exiting around Harajuku. The walkable portion of the park isn't very large and could be done in 1-2 hours depending on how much time you stop to smell the roses (if there are any). As a comparison, it took me almost a full day to circle around and explore Central Park in New York.