The seminal indian history text to cover the period up to about 1300 AD is

• Early India: From the Origins to AD 1300 by Romila Thapar

It is a good all round reference to classical india. However, it goes into details of Northern India a lot more than Southern India. If you want to go into the details of Southern India, the classic text is

• A History of South India: From Prehistoric Times to the Fall of Vijayanagar by K.A.Nilakanta Shastri

(Fall of vijayanagar is 1565 AD)

The Shastri text is old and some parts are not accurate* but comprehensive so you get a good overall picture of the social, economic, political and religious conditions in south india.

*if I recall, the latest edition has some commentary by R. Champakalakshmi (who is an outstanding south indian historian herself) about how Shastri imputes a lot more centralization to the Chola empire than reality. You can check her (Champakalakshmi) books for lot more detail about south india.