Argumentation was established by the Nyaya school. In his book, A Critical Survey of Indian Philosophy (https://archive.org/details/IndianPhilosophyACriticalSurvey), the author writes at the beginning of his chapter on the Nyaya:
The sage Gotama is the founder of the Nyaya School. He is also known as Gautama and as Aksapada. Nyaya means argumentation and suggests that the system is predominately intellectual, analytic, logical and epistemological. It is called Tarkashastra or the science of reasoning; Pramanashastra or the science of logic and epistemology; Hetusvidya or the science of causes; Vadavidya or the science of debate; and Anviksiki or the science of critical study.
Gotama's Nyaya-sutra was commented upon by Vatsyayana in his Nyaya-Bhasya. On this Uddyotakara wrote his Varika which was commented upon by Vachaspati in his Tatparya-tika. Udayana's Nyaya-kusumanjali and Jayanta's Nyaya-manjari are the other important works of this school. THe Navya-nyaya or the modern school of Indian logic begins with the epoch-making Tattva-chintamani of Gangesha. Vasudeva, Raghynatha, Mathuranatha, Jagadsiha and Gadadhara are the eminent logicians of this school.
A reading of the chapter is a good introduction to the Nyaya school. There is also a translation by Swami Madhavananda of the Bhasa Pariccheda with Siddhanta Muktavali written by Visvanatha-Nyaya-Pancanana Bhattacarya in the 1700s AD which is a study on the Navya-nyaya.
As a side note, in most debates one person takes the Siddhanta, conclusions to be established. The other takes the Purvapaksha, objections raised.