Shinjuku Gyoen is one of the largest parks in Tokyo and, in my opinion, the best. It used to have links with the Japanese royal family but it is now open to the public with an admission charge of 200 yen.
The garden has three styles: English Landscape, French formal and Japanese classical.
The English landscape garden looks somewhat like Greenwich Park in London and is popular with families having picnics during spring and autumn. It does get quite crowded as it is the first area that you come to as you enter the main entrance gate. It has a nice view of Tokyo Tower. The French formal part also gets crowded.
My recommendation is to venture further into the Japanese classical section. The garden fuses old (bonsai trees and a traditional Japanese lake) with modern (the stunning views of the Shinjuku skyscrapers). This area is less crowded than the other parts and the most peaceful.
The place is packed in spring when the cherry blossoms are in bloom. I recommend visiting the park during this time as the atmosphere is truly magical. The skies are filled with the faint pink of the cherry blossoms whilst the ground is filled with locals eating sushi from their bento boxes.
The park is huge at 3.5 km, so you could easily spend a day exporing the various parts. You can get there by getting off at Shinjuku-gyoemmae Station and it is about 5 mins on foot from here.