Questions & Answers about После разговора с другом мне стало ещё спокойнее.
Because после always takes the genitive case.
- разговор = conversation, talk
- после разговора = after the conversation / after talking
So разговор changes to разговора because it is governed by после.
This is a very common pattern in Russian:
- после урока = after the lesson
- после работы = after work
- после фильма = after the film
Here с means with, and in that meaning it normally takes the instrumental case.
- друг = friend
- с другом = with a friend / with my friend
So the form другом is instrumental singular.
Be careful: с can also mean from/off, and then it may take a different case. But in this sentence it clearly means with, so instrumental is used.