There are different ways you can start a new conversation:
Go to the chat menu and tap on the chat icon, located in the toolbar, at the bottom of the screen.
Once there, you can start a new chat by tapping on the button located at the bottom right corner of the screen.
Moreover, active conversations with other users will be displayed in the chat menu. There, you'll be able to continue chatting with them.
You can start a chat from an ongoing game. To do so, tap on the chat icon located in the upper right corner of the screen.