The Hippodrome is located in the square facing the Sultan Ahmet Mosque. The square was a circus that was the sporting and social centre during the Byzantine Period. There of the important monuments of the Hippodromw are still standing. They are the Obelisk, the Serpentine Column, and the Walled Obelisk.
Where is the Obelisk, Serpentine Column, Walled Obelisk ?
Obelisk is from the Ayasofya end of the square the first monument.
The Walled Obelisk is across the square and between them stands the smaller bronze monument which is known as Serpentine Column.
The Walled Obelisk is the last of the extant monuments.
Google Maps - Sultan Ahmet Square - Hippodrome

Its construction was started by Septimus Severus in the beginning of the 3rd century. However, it was not completed until the reign of Constantine.
Obelisk was brought from Eygypt with great difficulty, and erected only after months of effort by an army of workers. Erecting the obelisk took a long time and so much efffort that the Emperor Theodosius, who succeeded in the task in 390.
Serpetine Column was originally erected in the Temple of Apollo at Delphi in the 5th century BC. it was relocated to center of hippodrome by Constantine I the Great in 324. The serpent heads of the column remained intact until the end of the 17th century. And then one of heads was found and now it is on display at the nearby Istanbul Archaeology Museums.
The Walled Obelisk exact construction date is unknown, but it is named after Constantine VII after he repaired it in the tenth century.

You can go Eminönü - Zeytinburnu Tram.
You get off Sultanahmet Station from Tram then you can see which mosque has six minarets.
You can reach from Eminönü or Beyazıt to walk.
İETT Public Buses - Which public buses to way Eminönü
İETT Public Buses - Which public buses to go Beyazıt

Sultan Ahmet Square - hippodrome serves as an entertainment center during festivities of Ramadan. İf you have oppurtunities,you should go to Sultan Ahmet Square - hippodrrome during of Ramadan like a festival area you enjoy too much.