The easiest way to add someone to your friends list is to view their profile and click the big green "Request Connection" button:
That person will then be sent a connection (friend) request, and has the option of either accepting it, or denying it. You will be notified of their choice after they do so, and they will show up in your friends list.
You can see people you've added to your friends list by clicking on the "Connections" menu, and then choosing "Manage Your Connections":
