It depends on your preferences and budget. If you would like to live near central, many major central areas such as Roppongi, Shinjuku, etc, have apartments for rent as well, but rents are high. If you like more quiet neighborhood, Koenji, Ogikubo, Sangenjaya, Nakano, Kitasenju are pretty popular districts. Generally the west side of Tokyo are more young and modern and east side of Tokyo are more historical and traditional.
Rent gets cheaper if you get close to Chiba Prefecture, Saitama Prefecture, and Kanagawa Prefecture.