The garden is a ten minute walk from Shinjuku Station. Take the South Exit, and after leaving the station, turn left and follow the main road. Eventually you will get to an outdoor escalator going down to the ground level. Take the escalator and continue straight, and you will see Shinjuku Gyoen ahead.
From Tokyo Station to Shinjuku Station:
Take the Chuo Line direct to Shinjuku Station (15 minutes, ¥200)