The fort houses the Diwan-e-am, the hall for public audience with the emperor, where he would hear public grievances etc, and the Diwan-e-khas, the hall of private audiences with the emperor.
These and other mughal era features are common to the Agra fort and the Lahore fort.
The fort has political significance in todays India, since the Prime minister addresses the nation from here on Independence day, on August 15th every year.
Take the underground metro service to Red Fort. The most central access points to the metro are probably Connaught Place and India Gate (Central Secretariat stations). Thats probably the best way to reach any of the Old Delhi areas these days.

