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You get your friend code when you connect to the internet for the first time.
When your about to launch you DS's WiFi feature the game will say that the system treats your game and your DS as a set and that you should always use the same system with same game when going online. You'll have your friend code after a successful connection.

Specifically for Animal Crossing you'll find your friend code when you click the key icon in the Friend Roster (yellow heart icon).

Now, to get a friend's friend code you need to have a way to talk to them normally (like actually talking to them, over the phone, forums...). Click "Register" in the Friend Roster and ask said friend what their friend code is and input that, filling in the other information like Name and Town. You'll also need to give your friend code to your friend.

After that, you (and your friend) are read to play!

By the way, this is the same for all the other Nintendo WiFi games (DS and Wii), if not slightly different.

No. The only characters that go in and out and care about the dedicated sections of the museum are the four player characters and their friends (aka: you and your friends).
Blathers may own the museum but he's really only there to take thing from you for the exhibits.