Recent activities are everything you do on Facebook, from commenting on a friend's wall to posting photos or changing your relationship status. If a friend visits your profile they can see all of your actions -- a humbling thought, eh? -- unless they don't have permission.
Put simply, the only way to prevent people from seeing "John Lonelypants is now single" on their News Feed is to change your privacy settings so that your current relationship is hidden from your friends. This in turn makes any relationship-related status changes invisible.
Likewise, the only way to prevent other people from seeing which things you 'like' is to prevent them from seeing all of your likes and interests. It's either one way or the other with Facebook's current privacy model.
To begin, click Account in the top right corner, and then Privacy Settings -- or click this link.