I want to make a friendship with some person, but I don't want to let some of my friends know that me and that person have become friends. How can I achieve this? What privacy settings changes do I need to make? Is there any way to do this?
Click on the little triangle on the far right side of blue bar and choose Activity Log from the menu. Then in the left sidebar under Comments click MORE and then Friends.
Under the blue bar at the top it will say "Who can see your friend list?" followed by some icons. Click the square-like icon and uncheck all the check boxes corresponding to places where you do not want your new friendship to appear.
If you don't even want people to see your new friend by looking at your friend list on your profile you can click on the icon before this to change who can see your friend list; make it visible to Only Me or choose Custom and exclude certain people from seeing your friend list.
Keep in mind that if your new friend has their friend list visible, other people may see your friendship as part of your new friend's activity or on your new friend's friend list, or in the list of Mutual Friends if they are also friends with the person.