Question

where did Buddha deliver his first sermon ? By what name is this event known ?

Answer 1

Buddha gave his first sermons to five monks at Sarnath it is called the by Dhammacakkappavattaba Sutta

Answer 2

At the time the sangha the community of the enlightened ones, was founded. The sermon Buddha gave to the five monks was his first sermon, called the Dhammacakkappavattana sutta. It was given on the full moon day of Asalha puja. Buddha subsequently also spent his first rainy season at sarnath at the Mulagandhakuti