To get new friends, you both need to register each other's friend codes. If you don't have a FC yet, you'll need to get one, and I don't remember how DX
Friends generally increase the likelihood of you pairing up in MP games, and allow you to meet up on games such as Animal Crossing where you already have to know the person.
Also, if you have friends and connect to Wi-Fi, there's a chance something will happen. For example, in Animal Crossing, if you've connected with a friend recently, there's a chance one of their villagers will move to your town. Also, people can do things like bring Katie the Kitten over when you exchange FCs.
Other than that, there's nothing you really need to know. Over time, you'll learn a lot about all this connecting stuff. XD