Instructions
Start the game you want to play with your friend as you normally would by placing the game disc in the Wii and then clicking on the "Game channel" on the Wii's main menu.
Find the game's online mode and then find the "My Friend code" link. This will also vary by game, but is usually prominently displayed on the game's menu screen. One prominent exception is "Animal Crossing," where you must enter the game and go to your town gate to find your friend code.
Contact your friend (through phone or instant message) and give him your friend code. He must also give you his friend code.
Register your friend. There should be a friend registration area near where you found your friend code. Simply put in your friend's name (the same way they put it in on their Wii) and then press the "Register" button. The Wii will search the network for your friend and once a match has been found, he will appear on your friends list.
Wait for your friend to add you, and when you see that both of you are online (and registered with each other)you will be able to enter a game together. There should be a "play online" function (this is the case with "Mario Kart and Super Smash Bros. Brawl") or the person may be added to your game automatically, as is the case with "Animal Crossing."