You put the @username (the @ is optional) in the search field then click on the user and send a message. Once a message is send the chat will obviously stay in the chat list. Same for public groups/channels.
Alternatively you can share/click a link with the format t.me/username to skip the search part.
I press "New Message" and get these options: "New Group", "New Contact", "New Channel".
If I press "New Contact" it asks for permission to access my contacts. If I refuse, it goes back to the three options mentioned above.
I've tried to find it for years, and I just have the other person message me first.