You can only send someone a friend request by sending them a message, and you can only send a message to another player when they're online. It's annoying but at least once they're on your list they stay on your list. Note that messages in your inbox are deleted when you go offline. Unfortunately, you can also only send messages/add friends when the person is in a city and not on a quest.

When you create a character, you are immediately put in Moga village in the single player form. If you want to jump straight to online, choose "Quit Game" and then select the option "Mode Select." When you are at the mode select screen, choose "City" (the option on the right).